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A THEORY OF ORDINARY P-ADIC CURVES A Theory of Ordinary p-adic Curves By Shinichi MOCHIZUKI* Table of Contents Table of Contents Introduction §0. Statement of Main Results §1. Review of the Complex Theory Beltrami Diﬀerentials The Beltrami Equation The Series Expansion of a Quasiconformal Function Uniformization of Hyperbolic Riemann Surfaces Uniformization of Moduli Stacks of Hyperbolic Riemann Surfaces Quasidisks and the Bers Embedding The Inﬁnitesimal Form of the Modular Uniformizations Coordinates of Degeneration The Parabolic Case Real Curves §2. Translation into the p-adic Case Gunning’s Theory of Indigenous Bundles The Canonical Coordinates Associated to a Kähler Metric The Weil-Petersson Metric from the Point of View of Indigenous Bundles The Philosophy of Kähler Metrics as Frobenius Liftings The Dictionary Loose Ends Received September 8, 1995. Revised October 2, 1996. 1991 Mathematics Subject Classiﬁcation: 14H10, 14F30 * Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan 1 Chapter I: Crystalline Projective Structures §0. Introduction §1. Schwarz Structures Notation and Basic Deﬁnitions First Properties of Schwarz Structures Crystalline Schwarz Structures and Monodromy Correspondence with P 1 -bundles Schwarz Structures and Square Diﬀerentials Normalized P 1 -bundles with Connection The Schwarzian Derivative §2. Indigenous Bundles Basic Deﬁnitions and Examples First Properties Existence and de Rham Cohomology Indigenous Bundles of Restrictable Type §3. The Obstruction to Global Intrinsicity Introduction of Cohomology Classes Computation of the Second Chern Class The Case of Dimension One Appendix: Relation to the Complex Analytic Case Chapter II: Indigenous Bundles in Characteristic p §0. Introduction §1. FL-Bundles Deformations and FL-Bundles The p-Curvature of an FL-Bundle §2. The Verschiebung on Indigenous Bundles The Deﬁnition of The Verschiebung The p-Curvature of an Admissible Indigenous Bundle The Inﬁnitesimal Verschiebung Diﬀerential Criterion for Admissibility 2 §3. Hyperbolically Ordinary Curves Basic Deﬁnitions The Totally Degenerate Case The Case of Elliptic Curves: The Parabolic Picture The Case of Elliptic Curves: The Hyperbolic Picture The Generic Uniformization Number Chapter III: Canonical Modular Frobenius Liftings §0. Introduction §1. Generalities on Ordinary Frobenius Liftings Basic Deﬁnitions The Uniformizing Galois Representation The Canonical p-divisible Group Logarithms of Periods Compatibility of Diﬀerentials Canonical Liftings of Points in Characteristic p Canonical Multiplicative Parameters Canonical Aﬃne Coordinates The Relationship Between Aﬃne and Multiplicative Parameters §2. Construction of the Canonical Frobenius Lifting Modular Frobenius Liftings Indigenous Sections of D Frobenius Invariant Indigenous Bundles §3. Applications of the Canonical Frobenius Liftings Canonical Liftings of Curves over Witt Vectors Canonical Aﬃne Coordinates on M g,r Topological Markings and Uniformization by Quadratic Diﬀerentials Canonical Multiplicative Parameters The Case of Elliptic Curves Chapter IV: Canonical Curves §0. Introduction 3 §1. The Canonical Galois Representation Construction and Global Properties The Horizontal Section over the Ordinary Locus The Canonical Frobenius Lifting over the Ordinary Locus §2. The Canonical Log p-divisible Group Log p-divisible Groups at Inﬁnity Construction of the Canonical Log p-divisible Group Review of the Theory of [Katz-Mazur] §3. The Compactiﬁed Canonical Frobenius Lifting The Canonical Frobenius Lifting and the Canonical Log p-divisible Group Local Analysis at Supersingular Points Global Hecke Correspondences §4. p-adic Green’s Functions Compactiﬁed Frobenius Liftings The Height of a Frobenius Lifting Admissible Frobenius Liftings Geometric Criterion for Canonicality Chapter V: Uniformizations of Ordinary Curves §0. Introduction §1. Crystalline Induction The Crystalline-Induced MF ∇ -object The Ring of Additive Periods The Crystalline-Induced Galois Representation Relation to the Canonical Aﬃne Coordinates The Parabolic Case §2. Canonical Objects over the Stack of Multiplicative Periods The Stack of Multiplicative Periods The Canonical Log p-divisible Group The Canonical Frobenius Lifting Bibliography Index 4 Introduction §0. Statement of Main Results The goal of this paper is to present a theory of r-pointed stable curves of genus g over p-adic schemes (for p odd), which, on the one hand, generalizes the Serre-Tate theory of ordinary elliptic curves to the hyperbolic case (i.e., 2g − 2 + r ≥ 1), and, on the other hand, generalizes the complex uniformization theory of hyperbolic Riemann surfaces (reviewed in §1 of this introductory Chapter) due to Ahlfors, Bers, et al. to the p-adic case. We begin by setting up the necessary algebraic machinery: that is, the language of indigenous bundles (due to Gunning, although we rephrase Gunning’s results in a more algebraic form). An indigenous bundle is a P 1 -bundle over a curve, together with a connection, that satisfy certain properties. One may think of an indigenous bundle as an algebraic way of encoding uniformization data for a curve. We then study the p-curvature of indigenous bundles in characteristic p, and show that a generic r-pointed stable curve of genus g has a ﬁnite, nonzero number of distinguished indigenous bundles (P, ∇ P ), which are characterized by the following two properties: (1) the p-curvature of (P, ∇ P ) is nilpotent; (2) the space of indigenous bundles with nilpotent p-curvature is étale over the moduli stack of curves at (P, ∇ P ). We call such (P, ∇ P ) nilpotent and ordinary, and we call curves ordinary if they admit at least one such nilpotent, ordinary indigenous bundle. If a curve is ordinary, then choos- ing any one of the ﬁnite number of nilpotent, ordinary indigenous bundles on the curve completely determines the “uniformization theory of the curve” – to be described in the following paragraphs. Because of this, we refer to this choice as the choice of a p-adic quasiconformal equivalence class to which the curve belongs. After studying various basic properties of ordinary curves and ordinary indigenous bundles in characteristic p, we then consider the p-adic theory. Let M g,r be the moduli stack of r-pointed stable curves of genus g over Z p . Then we show that there exists a ord canonical p-adic (nonempty) formal stack N g,r together with an étale morphism ord N g,r → M g,r ord such that modulo p, N g,r is the moduli stack of ordinary r-pointed curves of genus g, together with a choice of p-adic quasiconformal equivalence class. Moreover, the generic ord degree of N g,r over M g,r is > 1 (as long as 2g − 2 + r ≥ 1 and p is suﬃciently large). It is ord over N g,r that most of our theory will take place. Our ﬁrst main result is the following: 5 ord Theorem 0.1. Let C log → (N g,r ) log (where the “log” refers to canonical log stack struc- tures) be the tautological ordinary r-pointed stable curve of genus g. Then there exists a ord log , together with a canonical indigenous bun- canonical Frobenius lifting Φ log N on (N g,r ) log log dle (P, ∇ P ) on C . Moreover, Φ N and (P, ∇ P ) are uniquely characterized by the fact (P, ∇ P ) is “Frobenius invariant” (in some suitable sense) with respect to Φ log N . Moreover, there is an open p-adic formal substack C ord ⊆ C of “ordinary points” of ord the curve. The open formal substack C ord ⊆ C is dense in every ﬁber of C over N g,r . Also, there is a unique canonical Frobenius lifting log ord ) → (C log ) ord Φ log C : (C which is Φ log N -linear and compatible with the Hodge section of the canonical indigenous log bundle (P, ∇ P ). Finally, Φ log C and Φ N have various functoriality properties, such as func- toriality with respect to “log admissible coverings of C log ” and with respect to restriction to the boundary of M g,r . This Theorem is an amalgamation of Theorem 2.8 of Chapter III and Theorem 2.6 of Chapter V. In some sense all other results in this paper are formal consequences of the above Theorem. For instance, Corollary 0.2. The Frobenius lifting Φ log N allows one to deﬁne canonical aﬃne local coordinates on M g,r at an ordinary point α valued in k, a perfect ﬁeld of characteristic p. These coordinates are well-deﬁned as soon as one chooses a quasiconformal equivalence class to which α belongs. Also, at a point α ∈ M g,r (k) corresponding to a totally degenerate curve, Φ log N deﬁnes canonical multiplicative local coordinates. This Corollary follows from Chapter III, Theorem 3.8 and Deﬁnition 3.13. ord Let α ∈ N g,r (A), where A = W (k), the ring of Witt vectors with coeﬃcients in a ord perfect ﬁeld of characteristic p. If α corresponds to a morphism Spec(A) → N g,r which is Frobenius equivariant (with respect to the natural Frobenius on A and the Frobenius ord lifting Φ log N on N g,r ), then we call the curve corresponding to α canonical. Let K be the quotient ﬁeld of A. Let GL ± 2 (−) be the group scheme which is the quotient of GL 2 (−) by {±1}. ord Theorem 0.3. Once one ﬁxes a k-valued point α 0 of N g,r , there is a unique canonical ord α ∈ N g,r (A) that lifts α 0 . Moreover, if a curve X log → Spec(A) is canonical, it admits (1) A canonical dual crystalline (in the sense of [Falt], §2) Galois repre- sentation ρ : π 1 (X K ) → GL ± 2 (Z p ) (which satisﬁes certain properties); 6 (2) A canonical log p-divisible group G log (up to {±1}) on X log whose Tate module deﬁnes the representation ρ; log ord (3) A canonical Frobenius lifting Φ log ) → (X log ) ord over the X : (X ordinary locus (which satisﬁes certain properties). Moreover, if a lifting X log → Spec(A) of α 0 has any one of these objects (1) through (3) (satisfying various properties), then it is canonical. This Theorem results from Chapter III, Theorem 3.2, Corollary 3.4; Chapter IV, Theorem 1.1, Theorem 1.6, Deﬁnition 2.2, Proposition 2.3, Theorem 4.17. The case of curves with ordinary reduction modulo p which are not canonical is more ord complicated. Let us consider the universal case. Thus, let S log = (N g,r ) log ; let f log : X log → S log be the universal r-pointed stable curve of genus g. Let T log → S log be the ﬁnite covering (log étale in characteristic zero) which is the Frobenius lifting Φ log N of log log log Theorem 0.1. Let P → S be the inverse limit of the coverings of S which are log log log log log log iterates of the Frobenius lifting Φ log . Let X = X × T ; X = X × . log S S log P N T P We would like to consider the arithmetic fundamental groups def def Π 1 = π 1 ((X T log ) Q p ); Π ∞ = π 1 ((X P log ) Q p ) Unlike the case of canonical curves, we do not get a canonical Galois representation of Π 1 into GL ± 2 (Z p ). Instead, we have the following Theorem 0.4. There is a canonical Galois representation ρ ∞ : Π ∞ → GL ± 2 (Z p ) Moreover, the obstruction to extending ρ ∞ to Π 1 is nontrivial and is measured precisely by the extent to which the canonical aﬃne coordinates (of Corollary 0.2) are nonzero. Also, log there is a ring D S Gal with a continuous action of π 1 (T Q ) such that we have a canonical p dual crystalline representation Gal ρ 1 : Π 1 → GL ± 2 (D S ) (i.e., this is a twisted homomorphism, with respect to the action of Π 1 (acting through log )) on D S Gal ). Finally, the ring D S Gal has an augmentation D S Gal → Z p which is π 1 (T Q p Π ∞ -equivariant (for the trivial action on Z p ) and which is such that after restricting to Π ∞ , and base changing by means of this augmentation, ρ 1 reduces to ρ ∞ . 7 This follows from Chapter V, Theorems 1.4 and 1.7. All along, we note that when one specializes the theory to the case of elliptic curves, one recovers the familiar classical theory of Serre-Tate. For instance, the deﬁnitions of “ordinary curves” and “canonical liftings” specialize to the objects with the same names in Serre-Tate theory. The p-adic canonical coordinates on the moduli stack M g,r (Corollary 0.2) specialize to the Serre-Tate parameter. The Galois obstruction to extending ρ ∞ to a representation of Π 1 specializes to the obstruction to splitting the well-known exact sequence of Galois modules that the p-adic Tate module of an ordinary elliptic curve ﬁts into. For more detailed accounts of the results in each Chapter, we refer to the introductory sections at the beginnings of each of the Chapters. In the rest of this introductory Chapter, we explain the relationship between the p-adic case and the classically known complex case. Acknowledgements: I would like to thank Prof. Barry Mazur of Harvard University for providing the stimulating environment (during the Spring of 1994) in which this paper was written. Also, I would like to thank both Prof. Mazur and Prof. Yasutaka Ihara (of RIMS, Kyoto University) for their eﬀorts in assisting me to publish this paper, and for permitting me to hold lecture series at Harvard (Spring of 1994) and RIMS (Fall of 1994), respectively, during which I discussed the contents of this paper. Finally, I would like to thank Prof. Ihara for informing me of the theory of [Ih], [Ih2], [Ih3], and [Ih4]. This theory anticipates many aspects of the theory of the present paper (especially, the discussion of Frobenius liftings and pseudo-correspondences in Chapters III and IV). On the other hand, the techniques and point of view of Prof. Ihara’s theory diﬀer substantially from those of the present paper. Moreover, from a rigorous, mathematical point of view, the main results of Prof. Ihara’s theory neither imply nor are implied by the main results of the present paper. However, it is the author’s subjective opinion that philosophically, the motivation behind Prof. Ihara’s theory was much the same as that of the author’s. §1. Review of the Complex Theory In order to explain the meaning of the main results of this paper, it is ﬁrst necessary to review the complex theory of uniformization in a fashion that makes the generalization to ﬁnite primes more natural. This is the goal of the present Section. Since all of the material is “standard” and “well-known,” we shall, of course, omit proofs, instead citing references for major results. We shall say that a Riemann surface X is of ﬁnite type if it can be obtained by removing a ﬁnite number of points p 1 , . . . , p r from a compact Riemann surface Y of genus g. Note that in this case, Y and {p 1 , . . . , p r } are uniquely determined up to isomorphism. We shall say that the Riemann surface of ﬁnite type X is hyperbolic (respectively, parabolic; elliptic) if 2g − 2 + r ≥ 1 (respectively, 2g − 2 + r = 0; 2g − 2 + r < 0). In this paper, we shall be concerned exclusively with Riemann surfaces of ﬁnite type (and their uniformizations). This is because it is precisely these Riemann surfaces which correspond to algebraic objects. Also, we shall mainly be concerned with 8 the hyperbolic case, since this is the most diﬃcult. Indeed, from the point of view of the theory of uniformization and moduli, the elliptic case is completely trivial, and the parabolic case (although nontrivial) is relatively easy and explicit. In some sense, the theme of our review of the classical complex theory is that in most cases, there are two ways to approach results: the “classical” and the “quasiconformal.” Typically, the classical approach was known earlier, and is more geometric and intuitive. On the other hand, the classical approach has the drawback of producing theories and results that are only real analytic, rather than holomorphic in nature. By contrast the quasiconformal approach, which was pioneered by Ahlfors and Bers, tends to give rise to holomorphic structures and results naturally. It is thus natural that the connection between the “quasiconformal approach” and the p-adic theory should be much more natural and transparent. Beltrami Diﬀerentials Let X be a Riemann surface (not necessarily of ﬁnite type). Let us consider the complex line bundle τ X ⊗ ω X on X, where ω X is the complex conjugate bundle to the canonical bundle ω X , and τ X is the tangent bundle. Note that if s is a section of τ X ⊗ ω X over X, then we can consider its L ∞ -norm s ∞ , since the transition functions of τ X ⊗ ω X have complex absolute value 1. A Beltrami diﬀerential μ on X is a measurable section of the line bundle τ X ⊗ ω X such that μ ∞ < 1. Why the bundle τ X ⊗ ω X ? The reason is that this bundle is closely connected with the moduli of the Riemann surface X. Indeed, Let us consider an arbitrary C ∞ section μ of τ X ⊗ ω X . Now since τ X has the structure of a holomorphic line bundle, we have a ∂ operator on τ X . If we look at global C ∞ sections, this gives us a complex ∂ C ∞ (X, τ X ) −→ C ∞ (X, τ X ⊗ ω X ) which computes the analytic cohomology of τ X . If X is, for instance, compact, then this analytic cohomology coincides with the cohomology in the Zariski topology of the algebraic tangent bundle. Thus, for X compact and hyperbolic, the above complex has cohomology groups H 0 = 0, and H 1 = H 1 (X, τ X ), which is well-known to be the space of inﬁnitesimal deformations of X. Moreover, if X is compact of genus g ≥ 2, and M g is the moduli stack of curves of genus g, then H 1 (X, τ X ) is precisely the tangent space to M g at the point deﬁned by X. At any rate, (for X arbitrary) we have a natural surjection 9 C ∞ (X, τ X ⊗ ω X ) → H 1 (X, τ X ) Thus, the image of μ under this surjection deﬁnes an inﬁnitesimal deformation of the complex structure of X. This establishes the relationship between sections of τ X ⊗ ω X and the moduli of X. The reason for considering measurable, rather than just C ∞ , sec- tions is that it is easier to obtain solutions to a certain diﬀerential equation, the Beltrami equation, when one works in this greater generality. The Beltrami Equation Having established the relationship between sections of τ X ⊗ ω X and inﬁnitesimal deformations, we now would like to integrate – i.e., to “give a reciprocity law” – that assigns to a section μ of τ X ⊗ ω X not just an inﬁnitesimal deformation of X, but an actual new Riemann surface, i.e., a new complex structure on the topological manifold underlying X. To do this, we consider the Beltrami equation ∂f = μ · ∂f which we regard as a diﬀerential equation in the unknown function f . It is a nontrivial result (proven, for instance, in [Lehto2]) that when μ is a Beltrami diﬀerential, there exist local L 2 solutions f to the Beltrami equation that are homeomorphisms (where they are deﬁned). Such functions f are called quasiconformal (with dilatation μ). If f and g (deﬁned on some open set U ⊆ X) are both quasiconformal with the same dilatation μ, then it is easy to see that ∂ applied to f ◦ g −1 (in the distributional sense) is zero. That is, f = h ◦ g for some biholomorphic function h. Thus, up to composition with a biholomorphic function, quasiconformal solutions to the Beltrami equation are unique. With these observations, we can deﬁne a new complex structure on X associated to a Beltrami diﬀerential μ as follows. Let us call the resulting Riemann surface X μ . Thus, the underlying topological manifold of X μ is the same as that of X. On an open set U ⊆ X, we take a local quasiconformal function f of dilatation μ, and deﬁne it to be a holomorphic function on X μ . By the essential uniqueness of solutions to the Beltrami equation, everything is well-deﬁned, and so we obtain a new global Riemann surface X μ . Thus, the assignment μ → X μ is the fundamental “reciprocity law” that we are looking for. 10 The Series Expansion of a Quasiconformal Function In order to really understand the Beltrami equation, it is useful to look at the explicit representation of its solutions as series “in μ” (as in [Lehto], pp. 25-27). We begin by considering Cauchy’s integral formula: 1 f (z) = 2πi 1 f (ζ) dζ − ζ − z π ∂D ∂f (ζ) dξdη ζ − z D for a function f with L 1 derivatives on an open disk D in the complex plane. Thus, if f (and its L 1 derivatives) are deﬁned on all of C, and f (z) → 0 as z → ∞, then we obtain f (z) = T ∂f where T is the operator on C ∞ functions ω with compact support given by 1 (T ω)(z) = − π ω(ζ) dξdη ζ − z C Put another way, (from the point of view of the theory of pseudodiﬀerential operators) T is the parametrix for the elliptic diﬀerential operator ∂. If we deﬁne the Hilbert transfor- mation H by ω(ζ) dξdη (ζ − z) 2 C 1 (H ω)(z) = − π then we obtain that ∂T = H. Also, it can be shown that ∂ and ∂ commute with both T and H. Now let us suppose that μ is a Beltrami diﬀerential on C (say, with compact support), and that f is quasiconformal on C with dilatation μ. Then f is holomorphic at inﬁnity, and so, after normalization, in a neighborhood of inﬁnity, it looks like f (z) = z + b n z −n n≥1 for some b n ∈ C. Thus, f (z) − z goes to 0 as z → ∞, so we obtain that ∂f (z) = 1 + ∂{f (z) − z} = 1 + ∂T ∂{f (z) − z} = 1 + H∂f (z) 11 Thus, since ∂f = μ · ∂f , it follows that ∂f = μ + μ · H∂f This integral equation has the formal solution ∂f = (μ · H) i μ i≥0 which converges in L 2 because (1) it can be shown that H extends to an isometry L 2 → L 2 ; (2) since μ is a Beltrami diﬀerential, μ ∞ < 1 (which thus explains this part of the deﬁnition of a Beltrami diﬀerential). Thus, applying the operator T , we get the series solution f (z) = z + T { (μ · H) i μ} i≥0 to the Beltrami equation. From our point of view, this series solution has two important consequences. First of all, the set of all possible μ clearly form an open subset of a (rather large) complex vector space (i.e., the space of measurable sections of τ X ⊗ ω X ). Thus, relative to the complex structure of this complex vector space, the series solution makes it clear that f depends holomorphically on μ. Second, it computes the inﬁnitesimal change in f as μ varies to ﬁrst def order. Namely, this term is given by φ = T (μ). Note that ∂φ = μ It turns out that this result – that ∂ applied to the inﬁnitesimal change φ in the solution to the Beltrami equation gives us back μ – holds for arbitrary Beltrami diﬀerentials μ. (See, e.g., [Gard], p. 72). The reason why this observation is interesting is as follows. Suppose, for simplicity, that μ is C ∞ . Let U be an open covering of X such that the intersection of any ﬁnite collection of open sets in U is Stein. Then by considering the standard isomorphism between the Čech cohomology (with respect to U ) and the ∂-cohomology of τ X , it thus follows that the inﬁnitesimal deformation X ·μ (where is “small”) in the complex structure of X given by solving the Beltrami equation is precisely the same as the inﬁnitesimal deformation given by mapping μ to H 1 (X, τ X ) via the surjection 12 C ∞ (X, τ X ⊗ ω X ) → H 1 (X, τ X ) considered previously. This completes the justiﬁcation of the claim that the assignment μ → X μ is an “integrated version” of the “inﬁnitesimal reciprocity law” C ∞ (X, τ X ⊗ ω X ) → H 1 (X, τ X ) that follows just from the deﬁnition of the ∂-cohomology of τ X . Uniformization of Hyperbolic Riemann Surfaces be its universal covering space. Thus, Let X be a hyperbolic Riemann surface. Let X X inherits a natural complex structure from X. Then one of the most basic results in the ﬁeld is that we have an isomorphism of Riemann surfaces ∼ X = H where H is the upper half plane. By considering the covering transformations of H → X, we get a homomorphism (well-deﬁned up to conjugation) ρ : π 1 (X) → Aut(H) ⊆ PSL 2 (R) which we call the canonical representation of X. There are (at least) two ways to prove this result. The ﬁrst approach is the classical approach, and goes back to Koebe’s work in the early twentieth century. It involves There is an intrinsic, a priori deﬁnition of considering Green’s functions G(−, −) on X. Green’s functions, which is not important for us here. A posteriori, that is, once one knows ∼ that X = H, we can pull-back the hyperbolic metric dx 2 + dy 2 y 2 so that we obtain a hyperbolic distance function on X. Then G(x, y) (for on H to X, is given by the logarithm of the hyperbolic distance between x and y. One can x, y ∈ X) ﬁnd a detailed exposition of this approach in [FK]. The second approach (which is more relevant to the p-adic case) is the approach of Bers ([Bers]). Suppose that X is obtained by removing r points from a compact Riemann surface Y of genus g. Then one ﬁrst observes that there exists a Riemann surface X which is obtained by removing r points from a compact Riemann surface of genus g and is isomorphic to H. Then one constructs (from purely whose universal covering space X 13 elementary considerations) a quasiconformal homeomorphism X ∼ = X. This quasiconfor- mal homeomorphism deﬁnes a Beltrami diﬀerential μ on X , which we can pull back to ∼ X = H to obtain a Beltrami diﬀerential μ H on H. By reﬂection, one extends μ H to a Beltrami diﬀerential μ on C. Then we solve the Beltrami equation for μ on C so that we obtain a quasiconformal homeomorphism f : C → C μ which goes to inﬁnity at inﬁnity. Let Γ be the group of Möbius transformation of H over X . Thus, H/Γ ∼ deﬁned by the covering transformations of X = X . Then it follows from the uniqueness of solutions to the Beltrami equation that Γ = f ◦ Γ ◦ f −1 μ μ def forms a group of Möbius transformations on C. Moreover, from the reﬂection symmetry of μ , it follows that f preserves the real axis, and hence so does Γ. It thus follows that μ H/Γ is a Riemann surface of ﬁnite type, and, by the deﬁnition of μ, that H/Γ ∼ = X. This completes the proof. It turns out that it is this approach of uniformizing a single Riemann surface (for each g, r) and then “parallel transporting” the result over the rest of the moduli space that will carry over to the p-adic case. Uniformization of Moduli Stacks of Hyperbolic Riemann Surfaces Let M g,r be the moduli stack of r-pointed smooth algebraic curves of genus g over C. Let M g,r be its universal covering space. Then the problem of uniformization of moduli is g,r . From the point of view of the Beltrami equation, to give an explicit representation of M this amounts to ﬁnding a small, ﬁnite-dimensional subspace T of the space of Beltrami diﬀerentials μ such that the assignment μ → X μ deﬁnes a covering space map T → M g,r . We begin by ﬁxing a “base point” of M g,r , which corresponds to a hyperbolic Riemann surface X. Let M(X) be the space of Beltrami diﬀerentials on X. Let Q be the space of holomorphic quadratic diﬀerentials on X with at most simple poles at the punctured points. Then there are two approaches to deﬁning morphisms from open subsets of Q into spaces of Beltrami diﬀerentials. The ﬁrst approach is that of Teichmüller. In this approach, if φ ∈ Q, we deﬁne a norm def φ = |φ| X Let V ⊆ Q be the set of φ with φ ≤ 1. Then Teichmüller’s uniformization map, for (nonzero) φ ∈ V , is given by 14 def φ → μ φ = (φ) φ |φ| where φ ∈ Γ(X, ω ⊗2 X ) is the complex conjugate of φ. It is easy to see that μ φ deﬁnes a Beltrami diﬀerential on X. Thus, we get a morphism V → M(X). If we compose φ → μ φ with μ → X μ , we get a morphism V → M g,r The main result of Teichmüller theory (see, e.g., [Gard], Chapter 6) is that this morphism g,r . One advantage of this approach is that it admits induces an isomorphism of V onto M a very satisfying geometric interpretation in terms of a foliation on X induced by φ and deforming X into X μ φ by deforming a canonical coordinate arising from the foliation. The main disadvantage of this approach from our point of view, however, is that the morphism φ → μ φ is neither holomorphic nor anti-holomorphic. Thus, it seems hopeless to try to ﬁnd an algebraic version of Teichmüller’s map. On the other hand, Bers’ approach is as follows. Since we now know that X can be uniformized by the upper half plane, let v X be the hyperbolic volume element on X induced by the hyperbolic volume element v H = dx ∧ dy y 2 on the upper half plane. Let X c be the conjugate Riemann surface to X. That is, the underlying topological manifold of X c is the same as that of X, but the holomorphic functions on X c are exactly the anti-holomorphic functions on X. Suppose that φ ∈ Q. Then by conjugating the “input variable,” we obtain that φ deﬁnes a section φ c of ω ⊗2 X c . Now deﬁne def −2φ μ φ = c v X c Then for some appropriate (see [Gard], pp. 100-104) open set V ⊆ Q, this μ φ deﬁnes a Beltrami diﬀerential on X c . Integrating, we get a Riemann surface X μ c . Then the assignment φ → X μ c deﬁnes a morphism V → M c g,r where the superscript “c” denotes the conjugate complex manifold. This morphism induces c g,r ([Gard], p. 101). The important thing here is that the an isomorphism of V onto M correspondence φ → μ φ is holomorphic. Since μ → X μ c is always holomorphic, it thus 15 c is biholomorphic. Put another way, we have a follows that the isomorphism V ∼ = M g,r holomorphic embedding g,r → Q c B : M which is called the Bers embedding. This embedding will be central to our entire treatment of the complex theory, and its p-adic analogue will be central to our treatment of the p-adic theory. Quasidisks and the Bers Embedding One can also deﬁne the Bers embedding in terms of Bers’ simultaneous uniformization and Schwarzian derivatives. For details, see [Gard], pp. 100-101. To do this, we ﬁx an with H. Let H c be the lower half plane. Thus, if H uniformizes X, isomorphism of X c then H naturally uniformizes X c . Let Γ be the group of Möbius transformiations of C → X. Then we may think of the space which are the covering transformations for H = X M(X c ) of Beltrami diﬀerentials on X c as the space of Beltrami diﬀerentials on H c which are invariant under Γ. Let μ ∈ M(X c ). Let f μ : C → C be the unique quasiconformal homeomorphism which ﬁxes 0 and 1, goes to inﬁnity at inﬁnity, has Beltrami coeﬃcient μ on H c and is conformal on H. Let Γ μ = f μ ◦Γ◦(f μ ) −1 . Then it follows from the uniqueness of solutions to the Beltrami equation that Γ μ forms a group of Möbius transformations of C. Moreover, we have conformal isomorphisms f μ (H c )/Γ μ ∼ = X μ c ; f μ (H)/Γ μ ∼ = X It follows that if we take the Schwarzian derivative of the conformal “quasidisk” embedding f μ | H : H → C we get a Γ-invariant quadratic diﬀerential on H, hence a quadratic diﬀerential φ (with at most simple poles at the punctures) on X. The content of the Lemma of Ahlfors-Weill ([Gard], p. 100) is that the assignment: X μ c → φ c → Q. On the one hand, this description of the Bers embedding is is equal to B c : M g,r geometrically more satisfying than the deﬁnition given in the previous subsection, but it has the disadvantage that it obscures the relationship between the hyperbolic and parabolic cases. So far we have been mainly discussing the hyperbolic case, but we shall discuss the parabolic case later. 16 The Inﬁnitesimal Form of the Modular Uniformizations Often it is useful to express these modular uniformizations in their inﬁnitesimal form, as metrics. On the one hand, the global uniformizations can always be essentially recovered by integrating the metrics, and on the other hand, metrics, being local in nature, can often be studied more easily. In the Teichmüller case, if K is deﬁned by φ = K − 1 K +1 g,r , given by then one obtains a distance function on M d(X, X μ φ ) = 1 log(K) 2 which turns out to be equal to the general hyperbolic distance introduced by Kobayashi for an arbitrary hyperbolic complex manifold (see [Gard], Chapter 7, for an exposition). The inﬁnitesimal form of this distance is given by the norm φ = X |φ| on quadratic diﬀerentials (see [Royd]). We shall be more interested in the case of the Bers embedding g,r → Q c B : M By using the hyperbolic volume form v X on X, we obtain the Weil-Petersson inner product: def φ, ψ = X φ · ψ v X for φ, ψ ∈ Q. It is a result of Weil and Ahlfors that the resulting metric, called the Weil- Petersson metric on M g,r , is Kähler. Moreover, if we diﬀerentiate B, we get, at X, a map on tangent spaces dB : Q ∨ → Q c whose inverse is exactly the morphism Q c → Q ∨ deﬁned by the Weil-Petersson inner prod- uct. Finally, the coordinates obtained from the Bers embedding are canonical coordinates for the Weil-Petersson metric ([Royd]). (We shall review the general theory of canonical coordinates associated to a real analytic Kähler metric in §2.) It turns out that it is precisely the p-adic analogue of the Weil-Petersson metric that will play a central role in this paper. 17 Coordinates of Degeneration While the Bers coordinates are useful for understanding what happens in the interior of M g,r , they are not so useful for understanding what happens as one goes out to the boundary, that is, as the Riemann surface degenerates to a Riemann surface with nodes. To study this sort of degeneration, one ﬁxes a decomposition of the Riemann surface into “pants,” which are topologically equivalent to an open disk with two smaller disks in the interior removed. For a detailed description of the theory of pants and the coordinates they deﬁne, we refer to [Abikoﬀ], Chapter 2. In summary, what happens is the following. Let X be a hyperbolic Riemann surface (of genus g with r punctures), with a decomposition into pants. We shall call the curves on X which occur in the boundary of the pants partition curves. There are exactly 3g − 3 + r partition curves, α 1 , . . . , α 3g−3+r . We assume that this decomposition is “maximal” in the sense that each partition curve is a simple closed geodesic (in the hyperbolic metric on X). Then it turns out that the isomorphism class of X as a Riemann surface is completely determined by 3g − 3 + r complex numbers ζ i = l i e iθ (i = 1, . . . , 3g − 3 + r), one for each partition curve. Basically l i describes the circumference of the partition curve α i , while θ i describes the angle of twisting involved in gluing together the boundary curves of two neighboring pants to form α i . These coordinates ζ i are called the Fenchel-Nielsen coordinates of X. The degeneration corresponding to pinching α i to a node is given by l i → 0. This degeneration respects the hyperbolic metrics involved: that is, if a family of smooth X t degenerates to a nodal Riemann surface Z, then the hyperbolic metrics on the X t degenerate to the hyperbolic metric on Z (given by taking the hyperbolic metric on the smooth subsurface of Z which is the complement of the nodes). Thus, the Fenchel-Nielsen coordinates have the virtue of admitting a very satisfying diﬀerential-geometric description (as just summarized), but the disadvantage of not being holomorphic. On the other hand, one can deﬁne holomorphic coordinates (as in [Wolp]), as follows. Recall the quasidisk description of the Bers embedding. Thus, we had a μ ∈ M(X c ), and a quasiconformal homeomorphism f μ : C → C, together with a new group of Möbius transformations Γ μ . Then each α i deﬁnes (by integration) an element A i ∈ Γ μ . Up to conjugation, A i is of the form z → m i · z for some m i ∈ C with |m i | > 1. This complex number m i is uniquely deﬁned. Then the coordinates X μ c → (m 1 , . . . , m 3g−3+r ) are holomorphic in μ. In [Wolp], the relationship between these coordinates and the Bers coordinates is studied. In these coordinates, the degeneration of X μ c corresponding to the case where the partition curve α i is pinched to a node is given by m i → 1. It turns out that these coordinates are probably the best complex analogue to the “multiplicative parameters at inﬁnity” that we construct in the p-adic case. 18 The Parabolic Case So far we have mainly been discussing the case of hyperbolic Riemann surfaces, since this case is by far the most interesting. However, often it is very diﬃcult to make explicit computations for hyperbolic Riemann surfaces. Thus, in order to get one’s bearings, it is sometimes useful to consider the analogous constructions in the parabolic case, where explicit computations are much easier to carry out. Let X be a parabolic Riemann surface. Then X is either compact of genus 1, or it is isomorphic to the projective line minus two points. We shall mainly be interested in the compact case, where there are nontrivial moduli. Thus, let X be compact of genus 1. Then one can carry out Teichmüller theory in this case (as in [Lehto], Chapter V, §6). One can also deﬁne a parabolic analogue of the Bers embedding, as follows. Namely, we simply copy the formula def −2φ μ φ = c v X c of the hyperbolic case, except that we take v X c to be the parabolic volume element (as opposed to the hyperbolic volume element) on X c , with X c v X c = 1. Then one sees (as in [Lehto], p. 220) that one obtains a holomorphic embedding 1,0 → Q c B 1,0 : M whose image is an open disk D ⊆ Q c of some radius. One can also deﬁne a Weil-Petersson metric on M 1,0 by simply replacing the hyperbolic volume element used before by the parabolic volume element. A simple calculation then reveals that one obtains the standard hyperbolic metric on the open disk D. In particular, (just as in the hyperbolic case), the standard coordinate on D is normal at 0 for the Weil-Petersson metric. One thing that is interesting about this parabolic case is that even though the complex analytic stacks M 1,0 and M 1,1 are isomorphic, the “Bers theory” diﬀers substantially in 1,1 is far from being an open disk. the two cases. For instance, the Bers embedding of M In fact, (as the author was told by C. McMullen) the boundary of this hyperbolic Bers embedding has lots of cusps. A computer-generated illustration of this boundary appears in [McM]. Also, it is not diﬃcult to show that the Weil-Petersson metrics are quite diﬀerent. This contrasts considerably with the “Teichmüller theory” of M 1,0 and M 1,1 : Indeed, since Teichmüller’s metric always coincides with Kobayashi’s intrinsic hyperbolic metric, it follows that the Teichmüller metrics of M 1,0 and M 1,1 coincide. Real Curves A Riemann surface X of ﬁnite type is called real if X ∼ = X c . In other words, this means that the C-valued point deﬁned by X in the algebraic stack (M g,r ) R (over Spec(R)) is, 19 in fact, deﬁned over R (up to perhaps reordering the marked points). Various interesting properties of real Riemann surfaces (related to uniformization theory) are studied in [Falt2]. Many of these properties are obtained by looking at various one-dimensional real analytic submanifolds of a real X. From our point of view, however, the notable fact about real hyperbolic Riemann surfaces X is the following. Let φ : X ∼ = X c be a holomorphic isomorphism. For simplicity, suppose that there exists a point x ∈ X such that φ(x) = x c , and that φ c ◦ φ = id X . Fix ∼ c ∼ an isomorphism X = H c . On the other hand, φ = H. This induces an isomorphism X c induces a holomorphic isomorphism φ : H → H . Let C : H c → H be the conjugation Thus, ψ is an anti-holomorphic automorphism of H. Now let map. Let ψ = C ◦ φ. Π C = π 1 (X, x). Since X c has the same underlying topological space as X, we have Π C = π 1 (X c , x c ). Thus, φ induces an automorphism φ Π of Π C of degree 2. Let Π R be the extension 1 → Π C → Π R → Gal(C/R) → 1 which is the crossed product of Π C with Gal(C/R) given by letting the nontrivial element of Gal(C/R) act on Π C by means of φ Π . Now let us consider the Lie group def G(R) = {M ∈ GL 2 (R)| det(M ) = ±1}/{±1} Thus, PSL 2 (R) ⊆ G(R) ⊆ GL ± 2 (R), so we can write ρ C : Π C → GL ± 2 (R) for the canonical representation of X (uniformized by the upper half plane H). Note that a b the full group GL ± ∈ GL ± 2 (R) acts on the upper half plane as follows: if A = 2 (R), c d we let A(z) = aw + b cw + d where w = z (respectively, w = z) if det(A) is positive (respectively, negative). Thus, the map deﬁned by A is a holomorphic (respectively, anti-holomorphic) automorphism of H if det(A) is positive (respectively, negative). In particular, the anti-holomorphic automorphism ψ : H → H deﬁnes an element (which by abuse of notation we call) ψ ∈ G(R). Now note that if γ ∈ Π C , then ψ · ρ(γ) · ψ −1 = ρ(φ Π (γ)). Thus, by mapping the nontrivial element of Gal(C/R) in the crossed product deﬁnition of Π R to ψ, we see that we obtain a natural homomorphism ρ R : Π R → GL ± 2 (R) 20 which extends ρ C and is such that the composite with the determinant det : GL ± 2 (R) → R × is trivial on Π C and equal to the sign representation on Gal(C/R). It is this repre- sentation ρ R that will be relevant to our discussion of the p-adic case. §2. Translation into the p-adic Case In this Section, we discuss the dictionary for translating the complex analytic theory of §1 into the p-adic results discussed in §0. Undoubtedly, the most fundamental tool, which is, in fact, of an algebraic, not an arithmetic nature, is the systematic use of the indigenous bundles of [Gunning]. This enables one to get rid of the upper half plane, and thus to bring uniformization theory into a somewhat more algebraic setting. In any sort of nontrivial arithmetic theory of this nature, however, algebraic manipulations alone can never be enough. Thus, the fundamental arithmetic observation is the following: Kähler metrics in the complex case correspond to Frobenius actions in the p-adic case. Since one typically gets a natural Frobenius action for free modulo p, a Frobenius action typically means a canonical lifting of the natural Frobenius action modulo p. In fact, in some sense, if one sorts through the complex analytic theory reviewed §1, one can essentially distill everything down to two objects, both of which happen to be Kähler metrics: (1) the hyperbolic metric on a hyperbolic Riemann surface (which encodes the upper half plane uniformization); and (2) the Weil-Petersson metric on the moduli space (which encodes the Bers uniformization). Moreover, these two metrics are related to each other in the sense that the latter is essen- tially the push-forward of the former. In a similar way, the p-adic theory revolves around two fundamental Frobenius liftings: (1) the canonical Frobenius lifting on a canonical hyperbolic curve; and (2) the canonical Frobenius lifting on a certain stack which is étale over the moduli stack. The goal of this Section is to explain this analogy in greater detail. 21 Gunning’s Theory of Indigenous Bundles Let X be a compact hyperbolic Riemann surface. Let H → X be its uniformization by the upper half plane. Then by considering the covering transformations of H → X, we get a homomorphism (unique up to conjugation) ρ : π 1 (X) → Aut(H) ⊆ PSL 2 (R) which we call the canonical representation of X. If we regard ρ as deﬁning a morphism into PSL 2 (C), then we obtain (in the usual fashion), a local system of P 1 -bundles on X, which thus gives us a holomorphic P 1 -bundle with connection (P, ∇ P ) on X. By Serre’s GAGA, (P, ∇ P ) is necessarily algebraic. It turns out that P is always isomorphic to a certain P 1 -bundle of jets (which is also entirely algebraic). Thus, the upper half plane uniformization may be thought of as just being a special choice of connection ∇ P . A pair “like” (P, ∇ P ) (satisfying certain technical properties discussed in Chapter I, §2) is called an indigenous bundle. By working with log structures, one can also deﬁne indigenous bundles in a natural way for smooth X with punctures, as well as for nodal X. As emphasized earlier, the point of dealing with indigenous bundles is that they allow one to translate the upper half plane uniformization into the purely algebraic information of a connection on P . Of course, how one chooses this particular special connection on P is very nontrivial arithmetic issue. We shall call the pair (P, ∇ P ) consisting of P equipped with this particular connection the canonical indigenous bundle on X. Universally, over the moduli stack M g,r (of stable r-pointed curves of genus g over C), the space of all indigenous bundles forms a holomorphic torsor S g,r → M g,r over the logarithmic cotangent bundle Ω log M g,r /C of M g,r . In the holomorphic category, we shall see (in Chapter I, §3) that this torsor is highly nontrivial. In the real analytic category, however, the canonical indigenous bundle determines a trivializing section s H : S g,r → M g,r of this torsor. In fact, indigenous bundles also allow us to translate such diﬀerential-geometric in- formation as the hyperbolic geometry of X into algebraic terms. For instance, consider the degeneration of Riemann surfaces from the point of view of hyperbolic geometry. As reviewed in §1, this may be thought of in terms of certain geodesic partition curves whose lengths go to zero as a family of smooth X t degenerates to a nodal Riemann surface Z. From the complex theory, we know that the hyperbolic metric on X t degenerates to the hyperbolic metric on Z. Using indigenous bundles, we can translate this into a more al- gebraic statement as follows: We deﬁne the canonical indigenous bundle on Z to be the 22 indigenous bundle obtained by gluing together the canonical indigenous bundles of the pointed Riemann surfaces occurring in the normalization of Z. Then the statement is that as X t degenerates to Z, the canonical indigenous bundle on X t degenerates to the canon- ical indigenous bundle on Z. The statement that the lengths of the partition geodesics go to zero then takes the form that the monodromy of the limit indigenous bundle of the canonical indigenous bundles of the X t ’s is nilpotent at the nodes. The Canonical Coordinates Associated to a Kähler Metric In this subsection we discuss how a Kähler metric on a complex manifold can be used to deﬁne canonical aﬃne, holomorphic coordinates on the manifold locally in a neighborhood of a given point. We believe that what is discussed here is well-known, but our point of view is somewhat diﬀerent from that usually taken in the literature. Let M be a smooth complex manifold of complex dimension m. The complex analytic structure on M deﬁnes, in particular, a real analytic structure on M . Let μ be a real analytic (1, 1)-form on M that deﬁnes a Kähler metric on M . In particular, μ is a closed diﬀerential form. Let M c be the conjugate complex manifold to M : that is to say, we take M c to be that complex manifold which has the same underlying real analytic manifold structure as M , but whose holomorphic functions are the anti-holomorphic functions of M . Let us ﬁx a point e ∈ M . Let N be the germ of a complex manifold obtained by localizing the complex manifold M c × M at (e, e) ∈ M c × M (where this last expression makes sense since M c has the same underlying set as M ). Let Ω hol (respectively, Ω ant ) be the holomorphic vector bundle on N obtained by pulling back the bundle Ω M (respectively, Ω M c ) of holomorphic diﬀerentials on M (respectively, M c ) to M c × M via the projection M c × M → M (respectively, M c × M → M c ), and then restricting to N . Thus, in summary, we have a 2m-dimensional germ of a complex manifold N , together with two m-dimensional holomorphic vector bundles (locally free sheaves) Ω hol and Ω ant on N . Note that locally at e ∈ M , the fact that μ is real analytic means that we can write μ as a convergent power series in holomorphic and anti-holomorphic local coordinates at e. In other words, if we restrict μ to N , we may regard μ| N as deﬁning a holomorphic section of Ω hol ⊗ O N Ω ant (where O N is the sheaf of holomorphic functions on N ). Let d hol (respectively, d ant ) be the exterior derivative on N with respect to the variables coming from M (respectively, M c ). Note that since Ω hol is constructed via pull-back from M , we can apply d ant to sections of Ω hol . We thus obtain a sort of de Rham complex with respect to d ant : 0 −→ Ω hol d ant −→ Ω hol ⊗ O N Ω ant d ant −→ Ω hol ⊗ O N (∧ 2 Ω ant ) d ant −→ ... Relative to this complex, the section μ| N of Ω hol ⊗ Ω ant satisﬁes d ant μ| N = 0 (since μ is a closed form). It thus follows from the Poincaré Lemma that there exists a (holomorphic) section α of Ω hol that vanishes at (e, e) ∈ N and satisﬁes d ant α = μ| N . Let M e be the germ of a complex manifold obtained by localizing M at e ∈ M . Let 23 ι : M e c → N be the inclusion induced by the map M c → M c ×M that takes f ∈ M c to (f, e) ∈ M c ×M . Then ι ∗ (α) deﬁnes a holomorphic morphism β : M e c → Ω M,e , where Ω M,e is the aﬃne complex analytic space deﬁned by the cotangent space of M at e. Note, moreover, that although α (as chosen above) is not unique, β is nonetheless independent of the choice of α. Moreover, β is an immersion: Indeed, to see this, it suﬃces to check that the map induced by β on tangent spaces is an isomorphism, but this follows from the fact that d ant α = μ| N , and the fact that the Hermitian form deﬁned by μ is nondegenerate. In summary, we see that from the Kähler metric μ, we obtain a canonical holomorphic local aﬃne uniformization β c : M e → Ω c M,e Pulling back the standard aﬃne coordinates on Ω c M,e gives us a canonical collection of holomorphic coordinates on M e . Deﬁnition 2.1. We shall refer to these coordinates as the canonical holomorphic local coordinates of the Kähler manifold (M, μ) at e. We shall refer to β c as the canonical local aﬃne uniformization of the Kähler manifold (M, μ) at e. Now let us consider some basic well-known examples: Example 1. Let M = {z ∈ C| |z| < 1}, with the standard hyperbolic metric √ 2dz∧dz . 1−(z·z) hol Then z is a canonical coordinate at 0. Indeed, to see this it suﬃces to note that d (z·dz) = dz ∧ dz, which is equal to the metric modulo the ideal generated by z in O N . Note that by the Köbe uniformization theorem, this example essentially covers all hyperbolic Riemann surfaces. Example 2. Let M be the Teichmüller space of Riemann surfaces of genus g with r punctures, where 2g − 2 + r ≥ 1. Then as stated earlier, it is known ([Royd]) that the coordinates arising from the Bers embedding are canonical coordinates with respect to the Weil-Petersson metric on M . In fact, in this case, by Theorem 2.3 (proven below) the real analytic section s H deﬁned by the canonical indigenous bundle essentially already serves as an “α” in the above discussion. Thus, in a very real sense, the section s H already is the Bers embedding. 24 The Weil-Petersson Metric from the Point of View of Indigenous Bundles Let X be a compact hyperbolic Riemann surface. Let (π : P → X, ∇ P ) be the canonical indigenous bundle on X. Let Ad(P ) = π ∗ τ P/X be the push-forward of the relative tangent bundle of π. Thus, Ad(P ) is a rank 3 vector bundle on X, equipped with a simple Lie algebra structure, hence with a nondegenerate Killing form < −, − >: Ad(P ) ⊗ O X Ad(P ) → O X . Moreover, ∇ P induces a connection ∇ Ad on Ad(P ). Moreover, as an indigenous bundle, Ad(P ) comes equipped with a section σ : X → P (the “Hodge section”) which deﬁnes a Hodge ﬁltration F · (Ad(P )) on Ad(P ). (See Chapter I for more 1 (Ad(P ), ∇ Ad ) module details.) At any rate, we can take the ﬁrst de Rham cohomology H DR of (Ad(P ), ∇ Ad ). The Hodge ﬁltration on Ad(P ) then deﬁnes a Hodge ﬁltration on the de Rham cohomology, hence an exact sequence: ⊗2 1 0 → H 0 (X, ω X ) → H DR (Ad(P ), ∇ Ad ) → H 1 (X, τ X ) → 0 On the other hand, recall the representation that we used to deﬁne (P, ∇ P ): ρ : π 1 (X) → Aut(H) ⊆ PSL 2 (R) Let Ad(V R ) denote the π 1 (X)-module obtained by letting π 1 (X) act on the Lie algebra def sl 2 (R) by applying ρ and then conjugating matrices. Let Ad(V C ) = Ad(V R ) ⊗ R C. Then (it is elementary that) we have a “comparison theorem” that gives a natural isomorphism between the de Rham cohomology module just considered and the group cohomology of Ad(V C ): 1 H DR (Ad(P ), ∇ Ad ) ∼ = H 1 (π 1 (X), Ad(V C )) On the other hand, we also have: H 1 (π 1 (X), Ad(V C )) ∼ = H 1 (π 1 (X), Ad(V R )) ⊗ R C which, combined with the above comparison theorem, thus gives a real structure on 1 H DR (Ad(P ), ∇ Ad ). One way to express this real structure is as an R-linear conjugation 1 1 (Ad(P ), ∇ Ad ) → H DR (Ad(P ), ∇ Ad ). morphism (read: “Frobenius action”) c DR : H DR Now let us consider the relationship between c DR and the Hodge ﬁltration. If we ⊗2 1 compose the natural inclusion H 0 (X, ω X ) → H DR (Ad(P ), ∇ Ad ) with c DR followed by the 1 1 natural projection H DR (Ad(P ), ∇ Ad ) → H (X, τ X ), we obtain a C-bilinear form β : H 0 (X, ω ⊗2 ) ⊗ C H 0 (X, ω ⊗2 ) c → C (where the superscript “c” stands for the complex conjugate C-vector space). 25 Proposition 2.2. The form β is precisely the Weil-Petersson metric on quadratic diﬀer- entials deﬁned in §1 by means of integration. In particular, β is nondegenerate. Proof. In order to obtain β, we implicitly used the special case of Serre duality given by ⊗2 ). But in the complex analytic context, the pairing that deﬁnes H 1 (X, τ X ) ∼ = H 0 (X, ω X this sort of duality is given by integrating the product of ((0, 1)− and (1, 0)−) forms. The volume form v X appears for the sake of deﬁning the duality between ω X and ω X . With these remarks, the claim of the Lemma becomes a tautology. Now let us recall the real analytic section s H : M g,r → S g,r . Since S g,r → M g,r is a holomorphic torsor, we may form ∂s H , which gives a section of Ω log M g,r /C log ⊗Ω M g,r /C . On the other hand, the Weil-Petersson metric also deﬁnes a section μ WP of Ω log M g,r log ⊗ Ω M g,r /C . /C Now we have the following result (stated in [ZT], but from a somewhat diﬀerent point of view): Theorem 2.3. The form ∂s H is equal to μ WP . Proof. By introducing log structures, one can handle the general case; here, for simplicity, we restrict our attention to the case of smooth compact Riemann surfaces. Let us consider ⊗2 1 ) → H DR (Ad(P ), ∇ Ad ) with c DR followed the composite of the natural inclusion H 0 (X, ω X 1 1 by the natural projection H DR (Ad(P ), ∇ Ad ) → H (X, τ X ); this composite gives a C-linear morphism: ⊗2 ) → H 1 (X, τ X ) c H 0 (X, ω X which is invertible by Lemma 2.2. Taking its inverse, and dualizing, we obtain an element ⊗2 ⊗2 c δ ∈ H 0 (X, ω X ) ⊗ C H 0 (X, ω X ) On the other hand, sorting through the deﬁnitions, it is a tautology in linear algebra that the value of ∂s H at the point [X] ∈ M g is given by δ. But, combining this with Lemma 2.2, we see that we have proven the Theorem. The important point here is that this Theorem shows that: The Weil-Petersson metric, and hence the Bers embedding, is obtained precisely by considering the extent to which “Frobenius” – i.e., complex conjugation – is compatible with the canonical indigenous bundle section s H . 26 Stated in this way, the classical complex theory becomes all the more formally analogous to the p-adic theory to be discussed in this paper. The Philosophy of Kähler Metrics as Frobenius Liftings Before going into a detailed account of the correspondence between complex and p- adic results, we pause to explain some of the motivation for considering Kähler metrics as Frobenius liftings. Let S be a smooth p-adic formal scheme over Z p . A Frobenius lifting on S is a morphism Φ : S → S whose reduction modulo p is equal to the Frobenius morphism in characteristic p. Then the main point of the analogy is that just as (real analytic) Kähler metrics deﬁne canonical coordinates (as discussed above), Frobenius liftings Φ : S → S (that satisfy a certain technical condition called ordinariness – see Chapter III, §1 for details) also deﬁne canonical coordinates, as follows: The most basic example of an ordinary Frobenius lifting is the case when S is the p-adic completion of Z p [T, T −1 ] (where T is an indeterminate), and Φ −1 (T ) = T p . Then the theory of ordinary Frobenius liftings (Chapter III, §1) states that by means of a certain “integration” procedure, every ordinary Frobenius lifting on an arbitrary S becomes (after completing at a point of S) isomorphic to a product of copies of this basic example. This “integration procedure” is thus analogous to the integration procedure just reviewed which allowed us to construct canonical coordinates associated to real analytic Kähler metrics. The Dictionary The fundamental “nuts and bolts” of the complex theory lies in the Beltrami equa- tion. Suppose that we think of the Beltrami equation not as a diﬀerential equation whose unknown is the quasiconformal function f μ , but instead as an equation whose unknown is the conformal quasidisk embedding function f μ | H (in the discussion of quasidisks). A quasidisk embedding of the universal covering space of a hyperbolic Riemann surface X deﬁnes an indigenous bundle (P, ∇ P ) μ on X in a natural way. Thus, from this point of view, we can think of the Beltrami equation as an equation whose unknown is (P, ∇ P ) μ . Moreover, the Beltrami coeﬃcient μ deﬁnes the “shearing” or distortion factor between z and z. Thus, in summary, we may regard the Beltrami equation as an equation in the unknown (P, ∇ P ) μ in terms of the distortion factor (eﬀected by the quasidisk embedding f μ | H ) between z and its “Frobenius conjugate” z. On the other hand, the “nuts and bolts” of the p-adic theory lies in the study of the Verschiebung on indigenous bundles, which occupies most of Chapter II. As a function on the indigenous bundles of a hyperbolic curve in characteristic p, the Verschiebung – which is essentially the determinant of the p-curvature – measures the distortion factor between applying Frobenius to an inﬁnitesimal on the curve and applying Frobenius to an inﬁnitesimal motion in the (“quasidisk”) uniformization deﬁned by the indigenous bundle. Thus, for instance, when the p-curvature is nilpotent, there is no distortion factor, and so the indigenous bundle provides the “right” uniformization for the curve. In this sense, we 27 feel that there is an analogy between the Beltrami equation in the complex theory and the Verschiebung on indigenous bundles in the p-adic theory. Relative to this analogy, the fundamental existence and uniqueness theorem for solu- tions to the Beltrami equation becomes the result (in Chapter II) that the Verschiebung on indigenous bundles is ﬁnite and ﬂat. Since in the p-adic case, its degree is not one, we only have uniqueness up to a ﬁnite number of possibilities. This is why we get several distinct “quasiconformal equivalence classes” in the p-adic case. Moreover, the important integral operator “T ” – i.e., the parametrix to ∂ – which gives the ﬁrst term in the series expansion for f μ may be regarded as having its analogue in the p-adic theory in the inﬁnitesimal Verschiebung, which plays an important role throughout the paper. More obvious is the analogy between the canonical representation ρ C : π 1 (X) → PSL 2 (R) of a hyperbolic Riemann surface (arising from the upper half plane uniformiza- tion), and the canonical representation ρ ∞ : Π ∞ → GL ± 2 (Z p ) of an ordinary p-adic curve (in Theorem 0.4). Of course in the p-adic case, Π ∞ has a substantial arithmetic part in addition to its geometric part. Although generally in the complex case, there is not much of a Galois group to work with, at least for real curves, we saw at the end of §1, that one does get a natural representation ρ R of the full “arithmetic fundamental group” Π R into GL ± 2 (R). Moreover, our approach to constructing ρ ∞ in the p-adic case is very much akin to Bers’ approach to constructing ρ C in the complex case: Namely, if one traces through the proof (which lies in Chapters II through V), one sees that eﬀectively what we are doing is noting that the result is true for totally degenerate curves, and then transporting this result over the rest of the moduli stack of ordinary curves. Next let us consider metrics and geometry. As we stated earlier, in some sense, one can summarize the entire complex theory by saying: We start with the hyperbolic (Kähler) metric on a hyperbolic curve, deﬁne the Weil-Petersson (Kähler) metric on the moduli stack precisely so as to be compatible with the hyperbolic metric on the curves being parametrized; then our holomorphic uniformizations – i.e., both the upper half plane uniformization of the hyperbolic curve and the Bers uniformization of the moduli stack – are obtained by “integrating” the respective metrics. Similarly, the fundamental result in the p-adic theory – namely, Theorem 0.1 – is a result about the existence of certain Frobenius liftings on the universal hyperbolic curve and its moduli stack which are uniquely characterized by the fact that they are compatible with each other. Here the compatibility is expressed through the tool of the canonical indigenous bundle. Then, by “integrating” these Frobenius actions, we obtain canonical (p-adically holomorphic) coordinates (as in ord Corollary 0.2) on C ord and N g,r . This particular analogy lies at the heart of this work. The Bers coordinates and the coordinates of Chapter III, Theorem 2.4, are appropriate in the locus M g,r of smooth curves. For totally degenerate curves, one has multiplicative parameters (Chapter III, Deﬁnition 2.7) which we believe are analogous to the holomorphic coordinates of degeneration of [Wolp], reviewed in §1. For instance, both sets of parameters are holomorphic and naturally indexed by the nodes of the totally degenerate curve. For elliptic curves – regarded parabolically – one has, on the one hand, the well-known theory of the hyperbolic upper half plane, or unit disk, in the complex case, and Serre- 28 Tate theory in the p-adic case. It is interesting to note that at both types of primes (complex and p-adic), the parabolic theory may be obtained in a very precise sense as the parabolic specializations, respectively, of Bers’ theory and of the hyperbolic p-adic theory developed in this paper. In fact, this is one of our reasons for feeling that the canonical p-adic coordinates of Corollary 0.2 are the p-adic analogue, not of Teichmüller’s coordinates, but of Bers’: Namely, in addition to the fact that Teichmüller’s coordinates are not holomorphic, whereas Bers’ are, Teichmüller obtains the same coordinates for 1-pointed curves of genus 1 and parabolic elliptic curves. On the other hand, it is well-known that Bers’ coordinates are very diﬀerent for 1-pointed curves of genus 1 and parabolic elliptic curves, which is consistent with the fact that the canonical coordinates of Corollary 0.2 are also very diﬀerent for 1-pointed curves of genus 1 and parabolic elliptic curves. Loose Ends We close by saying that although, as described above, there are (what the author believes to be) very strong analogies between Bers’ complex theory and the p-adic theory presented here, the picture is by no means complete. For instance, one fundamental fact in the complex case is that all r-pointed smooth curves of genus g are quasiconformally equivalent, whereas in the p-adic case, the theory behaves as though there are several diﬀerent quasiconformal equivalence classes that are permuted around to each other by a certain monodromy action in such a way that there seems to be no one quasiconformal equivalence class “which is better than the others.” Ideally, one would like to have a much more complete understanding of this phenomenon. In particular, one would like to know precisely how many quasiconformal equivalence classes there are (at least generically), as well as a more explicit description of the set of such classes. Also, I still do not understand what the complex, or global, analogue of a “canoni- cal p-adic curve” is. For ordinary elliptic curves, since Serre-Tate canonical liftings have complex multiplication, one can ask what the hyperbolic analogue of having complex mul- tiplication is. Since having complex multiplication for an elliptic curve means having lots of isogenies, it is natural to ask if the proper hyperbolic analogue is having lots of cor- respondences, which are a sort of higher genus version of isogenies. If a hyperbolic curve does have a lot of correspondences, then one knows ([Marg]) that the image of its canonical representation is arithmetic. In Chapter IV, we prove that a canonical curve has lots of “pseudo-correspondences,” but unfortunately, at the time of writing, I do not see how to make these pseudo-correspondences into genuine correspondences, so that one could apply Margulis’ result. Another issue that arises in this connection is the question of whether one can characterize “hyperbolic curves with complex multiplication” – whatever the correct deﬁnition should be for this term – in terms of the Bers coordinates. Finally, as the title implies, the present work deals exclusively with the case of ordinary curves. In a complete theory, one would like to know what happens when one has a nilpotent indigenous bundle which is not ordinary. 29 At any rate, in summary, with respect to these three issues of quasiconformal equiv- alence classes, canonical curves, and non-ordinary curves, much work remains to be done. We hope to be able to address these issues in future papers. 30 Chapter I: Crystalline Projective Structures §0. Introduction The purpose of this Chapter is to study the algebraic analogue of projective structures on a Riemann surface. In particular, we prove many of the analogues of results of [Gunning] in a purely algebraic framework, often making use of the crystalline site where complex analytically one would restrict to a simply connected neighborhood on which one can integrate. Unlike Gunning, we make systematic use of the log structures of [Kato], which enable us to work with a very general sort of “log-curve,” that is, we can handle the case of curves with marked points, as well as singular nodal curves on an equal footing to the smooth case. In §1, we discuss the notion of a Schwarz structure, which is the algebraic analogue of [Gunning]’s projective structures. We relate Schwarz structures to projective bundles with connections as well as to square diﬀerentials, and we show that Schwarz structures naturally give rise to a Schwarzian derivative. (Moreover, in the Appendix to this Chapter, we show that for P 1 , this abstract notion of a Schwarzian derivative essentially coincides with the classical Schwarzian derivative.) The characterizing feature of §1 is that everything takes place locally on the curve in question. In §2, we discuss indigenous bundles (the direct algebraic analogue of [Gunning]’s indigenous bundles). What distinguishes §2 from §1 is that in §2, we work mainly over stable curves, and thus global issues on the curve come into play. In §2, we are still working locally, however, on the base. In §3, we perform various intersection theory calculations that allow us to prove that in most cases, there do exist any canonical indigenous bundles on the universal smooth curve over a moduli stack. Thus, in §3, we are concerned with issues that are global not only on the curve, but also on the base. It should be said that all the material in this Chapter is, in some sense, “well-known,” but I do not know of any modern reference that does things from this point of view. In particular, all the references that I know of (with the exception of [Ih], which is algebraic, but somewhat diﬀerent in point of view) discuss things only in the complex analytic case, and often work with “h αβ ’s” (i.e., cocycle classes) rather than with objects that have an intrinsic meaning. §1. Schwarz Structures In this Section, we introduce the crystalline analogue of what Gunning calls “projective structures on a Riemann surface.” (We shall call them Schwarz structures (after the Schwarzian derivative) to distinguish them from the analytic notion.) We begin by letting S be a connected noetherian scheme. Often, we shall prove results about arbitrary stable curves by working on various compactiﬁed moduli stacks. Thus, even if one is ultimately interested only in smooth curves, for certain proofs, we shall see that it is useful to develop 31 the machinery for arbitrary stable curves. To deal with singular curves, we shall use the theory of log schemes of [Kato]. Thus, we assume that S has a given ﬁne ([Kato], §2) log structure, and denote the resulting log scheme by S log . Notation and Basic Deﬁnitions Deﬁnition 1.1. Let f log : U log → S log be a morphism of log schemes whose underlying morphism of schemes f : U → S is of ﬁnite type, ﬂat and of relative dimension one. Then we shall say that f log is locally stable of dimension one if, for every point u ∈ U , there exist étale morphisms T → S and V → U × S T , together with v ∈ V mapping to u ∈ U such that when we pull-back the log structure on S (respectively, U ) to T (respectively, V ) to obtain log schemes V log and T log , one of the following holds: (1) V → T is smooth, and V log = V × T T log (where V and T denote the log schemes with trivial log structure); or (2) V → T is smooth, and there exists a section s : T → V such that if we denote by V s the log scheme deﬁned by the relative divisor Im(s) on V , then V log = V s × T T log ; or (3) let Y = Spec(Z[t]); X = Y [x, y]/(xy − t) (where x, y, and t are indeter- minates) and endow Y (respectively, X) with the log structure arising from the divisor t = 0 (respectively, xy = 0), so we get a morphism X log → Y log of log schemes; then there exists a morphism of log schemes T log → Y log , together with a morphism ζ log : V log → T log × Y log X log such that the underlying scheme morphism ζ of ζ log is étale, and the log structure of V log on V is the pull-back via ζ of the log structure on T log × Y log X log . In case (1) (respectively, (2); (3)), we shall say that f is smooth and unmarked (respectively, marked; singular) at u. Note that if f log : U log → S log is locally stable of dimension one, then it is always log smooth ([Kato], §3). Also, note that by étale descent, the images in U of all the sections s as in Case (2) above form a divisor in U which is étale over S. We shall refer to this divisor as the divisor of marked points in U . Now let us suppose that there exists an odd prime p which is nilpotent on S. We also suppose that we are given a closed subscheme S 0 = V (I) ⊆ S, where the sheaf of ideals I has a divided power structure γ. We denote the log scheme S 0 × S S log (where S 0 and S denote the log schemes which are the respective schemes endowed with the trivial log structure) by S 0 log . Let f log : U log → S log be locally stable of dimension one. Then we shall call a section of D Δ (U log × S log U log ) (the PD-envelope of the diagonal, as in [Kato], 32 §5) a bianalytic function over U log . Note that the bianalytic functions form a sheaf, which we denote O U bi , on the étale site of U . Let O U denote the sheaf on the étale site of U given by considering ordinary functions. Then the two projections U × S U → U give rise to injections i L : O U → O U bi and i R : O U → O U bi whose images we shall call the left-sided (respectively, right-sided) bianalytic functions on U log . We shall also refer to right-sided bianalytic functions as constant bianalytic functions, or bianalytic constants. We denote tensor products of an O U -module F over O U with O U bi via i L (respectively, i R ) by writing F on the left (respectively, right). Finally, we have a multiplication morphism μ : O U bi → O U . We denote the ideal subsheaf of O U bi which is the kernel of μ by J . We shall say that a bianalytic function f over some étale V → U is a bianalytic uniformizer on V if f is, in fact, a section of J which generates the line bundle J /J [2] ∼ = ω U log /S log as an O U - module. Let O U bi be the completion of O U bi with respect to the divided powers J [i] , and let J ⊆ O U bi be the closure of J in O U bi . We shall call sections of O U bi biformal functions, and use similar terminology for biformal functions as we do for bianalytic functions. Occasionally, we shall also need to make use of trianalytic (respectively, triformal) functions, i.e., sections of D Δ (U log × S log U log × S log U log ) (respectively, its completion with respect to the divided powers of the diagonal ideal). We denote the sheaf of trianalytic functions (respectively, triformal) on the étale site of U by O U tr (respectively, O U tr ), and we have left, right, and middle injections j 1 , j 2 , j 3 : O U → O U tr , as well as injections j 12 , j 23 , j 13 : O U bi → O U tr . We shall apply similar terminology and notation to trianalytic or triformal functions to that applied already to bianalytic functions. In particular, we shall call trianalytic functions that are in the image of j 23 trianalytic constants. Deﬁnition 1.2. Let S ⊆ O U bi be a subsheaf in the category of sets. We shall call S a Schwarz (respectively, pre-Schwarz) structure on U log if étale locally on U (i.e., for some étale cover V → U ), S has the following form: there exists some biformal uniformizer z ∈ Γ(V, S) such that for every étale W → V , and every section f ∈ Γ(W, O U bi ), then f ∈ Γ(W, S) if and only if (respectively, implies that) f can be written étale locally (on W ) in the form (az + b)/(cz + d), where a, b, c, d are biformal constants and d is invertible. It is clear that if S ⊆ O U bi is a pre-Schwarz structure on U log , then S is contained in a unique Schwarz structure S a ⊆ O U bi on U log , which we refer to as the Schwarz structure associated to S. If S is a Schwarz structure, then we shall denote by S × ⊆ S (respectively, L S ) the subsheaf consisting locally of functions of the form (az +b)/(cz +d), where: (1) z is a b a biformal uniformizer belonging to S; (2) d is invertible; and (3) is an invertible c d S × . Thus S × , L S , matrix of biformal constants (respectively, b = 0). We let L × S = L S × and L × S are all pre-Schwarz structures. We shall call L S (respectively, L S ) the sheaf of biformal uniformizers (respectively, pseudo-uniformizers) of S. Let G → U be the group scheme PGL 2 , and let B ⊆ G be the subgroup scheme which is the standard Borel subgroup of PGL 2 , i.e., the image of the lower triangular matrices. 33 First Properties of Schwarz Structures Proposition 1.3. The subsheaf L × S ⊆ S consisting of biformal uniformizers of S forms a B-torsor B S → U . Proof. This follows immediately from the deﬁnition of a Schwarz structure. The action of a 0 B is given by associating to a biformal uniformizer z and a matrix (where a, c, d c d are biformal constants, and a, d are invertible) the biformal uniformizer az/(cz + d). Note that every B-torsor T → U naturally deﬁnes a P 1 -bundle with a given section (by taking the quotient of P 1 × U T modulo the diagonal action of B, where B acts on P 1 by means of aﬃne transformations that ﬁx zero; the section is the image of the zero section of P 1 ). We shall refer to the P 1 -bundle P S → U associated to B S → U as the P 1 -bundle associated to the Schwarz structure S. We denote by σ S : U → P S the natural section (arising from the fact that the structure group is B rather than G). Proposition 1.4. Let S be a Schwarz structure on U log . Then P S ∼ = P(J /J [3] ), and σ S ∗ τ P S /U ∼ = (J /J [2] ) ∨ ∼ = τ U log /S log . In particular, if U → S is proper, then the height of σ S with respect to τ P S /U is −deg(ω U log /S log ). Proof. One sees by construction (e.g., by writing out transition functions) that the sheaf of nonzero relative rational functions of relative degree one (as in [EGA IV], §20) for P S → U that vanish at σ is naturally isomorphic to L × S . Thus, by considering Taylor expansions out to second order terms, we get an isomorphism O P S (−σ S )/O P S (−3σ S ) ∼ = J /J [3] (here we use that p is odd). On the other hand, by multiplying and then taking the residue at σ, we obtain a natural duality between O P S (−σ S )/O P S (−3σ S ) and π ∗ ω P S /U (3σ S ), where π : P S → U is the natural projection. Also, note that via this duality, the ﬁltration induced by π ∗ ω P S /U (2σ S ) ⊆ π ∗ ω P S /U (3σ S ) on O P S (−σ S )/O P S (−3σ S ) is the ﬁltration deﬁned by the submodule O P S (−2σ S )/O P S (−3σ S ). Since P S is clearly naturally isomorphic to the projectivization of π ∗ ω P S /U (3σ S ), we thus obtain the result. Crystalline Schwarz Structures and Monodromy Let S be a Schwarz structure on U log . We would like to associate to S a subsheaf (in the category of sets) of O U tr , which we shall call S 12 as follows. We work locally. Thus, we assume that there exists a biformal uniformizer z ∈ Γ(U, S). We consider the triformal function z 12 deﬁned by j 12 (z), where j 12 is the natural map O U bi → O U tr given by inclusion on the ﬁrst two factors. Then we let S 12 be the sheaf of all functions which étale locally can be written in the form (az 12 + b)/(cz 12 + d), where a, b, c, d are triformal constants and d is invertible. Note that the deﬁnition of S 12 does not depend on the choice of z, 34 so everything glues together, and we obtain the subsheaf S 12 of O U tr over our original U . On the other hand, we also have a subsheaf S 13 of O U tr deﬁned in the same way as S 12 , except with the roles of 2 and 3 reversed. Deﬁnition 1.5. We shall say that the Schwarz structure S is a crystalline Schwarz structure on U log if the two subsheaves S 12 and S 13 of O U tr coincide. Let S be a crystalline Schwarz structure on U log . Then we shall say that S has nilpotent monodromy if for every marked point s : T → V (with V → U étale), there exists a biformal uniformizer z ∈ S(V ) and a section a ∈ ω U log /S log (V ) such that the image of (dz) − i R (a) (where “d” is the exterior derivative on the right) in O U bi ⊗ O U s ∗ ω U log /S log is zero. Remark. Of course, one may also phrase the deﬁnition of a crystalline Schwarz structure as follows. First, note that O U bi , together with its right-hand sided O U -algebra structure and standard logarithmic connection, forms a quasi-coherent crystal of algebras A on the crystalline site of U log /S log . Then a crystalline Schwarz structure is a subsheaf of the sheaf A on the crystalline site of U log /S log satisfying certain properties. Since this point of view is only formally diﬀerent from the point of view of Deﬁnition 1.5, we shall use these two points of view interchangeably in what follows. log Let us suppose that S is a Schwarz structure on U . Let H ⊆ G be the open a b subscheme consisting of matrices of the form , where d is invertible. Note that c d H is stable under the action by B from the right. Thus, we can take the quotient of H × U B S (by the diagonal action of B) to obtain a ﬁber bundle H S → U with ﬁbers locally isomorphic to H → U . Similarly, we also obtain a G-torsor G S → U . It now follows immediately from the deﬁnitions that the sheaf deﬁned on the étale site of U by H S is naturally isomorphic to S × . Thus, if we assume that the Schwarz structure S is crystalline, we see that we get a natural isomorphism between the two pull-backs of H S → U via i L , i R : O U → O U bi , i.e., we get a logarithmic connection ∇ H S on H S → U . By basic facts about ﬁber bundles, this gives a logarithmic connection ∇ G S on G S → U and a logarithmic connection ∇ P S on P S → U , as well. Thus, in summary, to every crystalline Schwarz structure S, we have associated a natural P 1 -bundle with section and logarithmic connection (P S → U ; σ S : U → P S ; ∇ P S ). Moreover, it follows from the deﬁnition of the connection ∇ P S that by diﬀerentiating σ S by means of ∇ P S , we get an isomorphism τ U log /S log ∼ = σ S ∗ τ P S /U , which is called the Kodaira-Spencer morphism. Indeed, to see that this morphism is, indeed, an isomorphism, it suﬃces to realize that if, locally on U , one takes a biformal uniformizer z, the diﬀerence j 12 (z) − j 13 (z) generates j 23 ( J )O U tr . In addition to the fact that the Kodaira-Spencer morphism is an isomorphism, the logarithmic connection ∇ P S has another special property: If S has nilpotent monodromy, then we can make more explicit the way in which this monodromy acts. Indeed, let us recall from [Kato], §6, that if s : S → U is any marked point, then there exists a unique subsheaf M s of (O U bi ⊗ O U ,s −1 O S )/O S which is isomorphic to O S and annihilated by the monodromy operator of the standard logarithmic connection on O U bi . (Locally, 35 this subsheaf is generated by log(1 − δ), where δ = 1 − ( 1⊗t t⊗1 ), and t is a local generator of the ideal deﬁning s.) This subsheaf M s thus deﬁnes a section q s : S → P(J /J [3] ) that lies over s. Then it follows from these observations, plus Proposition 1.4, that Proposition 1.6. If S is a crystalline Schwarz structure on U log with nilpotent mon- odromy, then under the isomorphism P S ∼ = P(J /J [3] ) of Proposition 1.4, q s is ﬁxed by the monodromy action on P S ∼ = P(J /J [3] ) at s. Correspondence with P 1 -bundles So far, from a crystalline Schwarz structure, we have constructed a P 1 -bundle with section and logarithmic connection (satisfying certain properties). We can go the other way, as well. Suppose we are given a P 1 -bundle with section and logarithmic connection (π : P → U ; σ : U → P ; ∇ P ) such that the Kodaira-Spencer morphism obtained by diﬀerentiating σ via ∇ P gives an isomorphism τ U log /S log ∼ = σ ∗ τ P/U . Let P L and P R denote the pull-backs of π : P → U via i L , i R : O U → O U bi , respectively. Then the connection ∇ P deﬁnes an O U bi -linear isomorphism Ξ : P L ∼ = P R . Thus, we have a commutative diagram: Ξ P L −→ ⏐ ⏐ L π U bi id −→ P R ⏐ ⏐ R π U bi Let σ L (respectively, σ R ) denote the result of base-changing σ via i L (respectively, i R ). Then by applying (σ L ) ∗ , we can pull-back functions on P L to biformal functions on U log . Let R denote the étale sheaf of degree ≤ 1 relative rational functions (as in [EGA IV], §20) on P relative to π : P → U (i.e., the divisor of poles is ﬂat of degree ≤ 1 over U ) that are regular in a neighborhood of the image of σ. Let α R : P R → P denote the natural projection. Then it is easy to see that Ξ −1 (α R ) −1 (R) deﬁnes a sheaf of functions on P L that are regular in a neighborhood of Im(σ L ), so we can consider the subsheaf S of biformal functions on U log which is the image of (σ L ) −1 Ξ −1 (α R ) −1 (R). One checks immediately that S deﬁnes a Schwarz structure, and, moreover, that since the connection ∇ P is necessarily integrable (since the dimension of U over S is one), S is automatically crystalline. Thus, in summary, we have the crystalline analogue of Theorem 2 of [Gunning]: Theorem 1.7. If f : U → S is as above, then there is a natural one-to-one correspondence between crystalline Schwarz structures on U log and isomorphism classes of P 1 -bundles with section and logarithmic connection (π : P → U ; σ : U → P ; ∇ P ) on U log whose associated Kodaira-Spencer morphism is an isomorphism. Moreover, under this correspondence, the crystalline Schwarz structures with nilpotent monodromy correpond precisely to the triples such that ∇ P has nilpotent monodromy at the marked points. 36 Proof. We have already deﬁned maps going in either direction. Thus, it suﬃces to see that these maps are inverse to each other. Now it is easy to see that if we start with a P 1 -bundle with section and logarithmic connection as above, construct the associated crystalline Schwarz structure S, and then from that the associated P 1 -bundle P S with section σ S and logarithmic connection ∇ P S , then we get back our original data. Thus, it suﬃces to show that the map that associates a P 1 -bundle with section and connection to a crystalline Schwarz structure is injective. Let S and S be crystalline Schwarz structures on U log . Suppose that we are given a horizontal isomorphism α between P S and P S that takes σ S to σ S . Then α induces an isomorphism α B of the B-torsors B S and B S . Let A be the O U -algebra O U bi via ∼ × the morphism i R . Since as sheaves with B-action, B S ∼ = L × S and B S = L S , we get an × × isomorphism α L : L × S → L S , which, by mapping a biformal uniformizer z ∈ Γ(U, L S ) (where U → X is étale) to the biformal uniformizer α L (z) ∈ Γ(U, L × S ) ⊆ Γ(U, A), deﬁnes an automorphism α A of the PD-O U -algebra A that preserves the augmentation μ : A → O U . Moreover, it follows from the horizontality of α that α A is horizontal with respect to the standard logarithmic connection on A. On the other hand, it is immediate that A does admit any nontrivial horizontal automorphisms (as a PD-O U -algebra) that preserve μ. Thus, α A is the identity, and hence, S and S must be the same subsheaf of A. This completes the proof of the ﬁrst statement. The last statement follows directly from the deﬁnitions. Schwarz Structures and Square Diﬀerentials We would like to use Theorem 1.7 to exhibit the space of Schwarz structures as a torsor over the square diﬀerentials. Let π : P → U be a P 1 -bundle. Then we shall denote by Ad(P ) the vector bundle on U (of rank three with trivial determinant) given by π ∗ τ P/U . When we consider marked points, it is not enough just to deal with P 1 -bundles; we must deal with P 1 -bundles equipped with parabolic structures, as in [Sesh]. Thus, if our divisor D of marked points is given by sections p 1 , . . . , p r : S → U , we make the following Deﬁnition 1.8. A P 1 -bundle with parabolic structure on U log is deﬁned to be a P 1 -bundle π : P → U , together with sections q i : S → P lying over p i . A rank two vector bundle with parabolic structure on U log is a rank two vector bundle E, together with a parabolic structure on P(E). Let (π : P → U ; q 1 , . . . , q r ) be a P 1 -bundle with parabolic structure on U log . Then we deﬁne the subsheaf Ad q (P ) ⊆ Ad(P ) to be the sheaf of sections that vanish at the q i ’s. We deﬁne Ad c (P ) ⊆ Ad q (P ) to be the subsheaf of sections that vanish to second order (in the relative coordinate for π) at the q i ’s. Suppose that we are given a section σ : U → P that avoids all the q i . Let L = σ ∗ ω P/U . Then Ad(P ) gets a ﬁltration 0 = F 2 (Ad(P )) ⊆ F 1 (Ad(P )) ⊆ F 0 (Ad(P )) ⊆ F −1 (Ad(P )) = Ad(P ) given by considering sections of τ P/U that vanish to ﬁrst or second order at σ. Thus, for Ad(P ), we have: 37 F 1 ∼ = L; F 0 /F 1 ∼ = O U ; F −1 /F 0 ∼ = L −1 This ﬁltration induces ﬁltrations on Ad q (P ) and Ad c (P ). The subquotients are easily seen to be the following: For Ad q (P ), we have: F 1 ∼ = L(−D); F 0 /F 1 ∼ = O U ; F −1 /F 0 ∼ = L −1 For Ad c (P ), we have: F 1 ∼ = L(−D); F 0 /F 1 ∼ = O U (−D); F −1 /F 0 ∼ = L −1 log Often, L ∼ = ω U/S . Thus, for computational purposes, it is convenient to note that log ω U/S (−D) is none other than the relative dualizing sheaf of the morphism f : U → S. Now let us assume that π : P → U is given by P(J /J [3] ), with the section σ given by J /J [3] → J /J [2] , and the q i given by the sections “q s ” deﬁned in the paragraph preceding Proposition 1.6. Let ∇ P be a logarithmic connection whose Kodaira-Spencer morphism at σ is the identity and whose monodromy at the marked points is nilpotent and ﬁxes the q i . (It is not diﬃcult to see that such ∇ P always exist étale locally on U .) Then any other such logarithmic connection ∇ P on P → U is given by adding to ∇ P a log . On the other hand, the quadruples (π; σ; q i ; ∇ P ) and section of F 0 (Ad c (P )) ⊗ O U ω U/S (π; σ; q i ; ∇ P ) are isomorphic if and only if ∇ P can be obtained from ∇ P by applying an automorphism α of (π; σ; q i ) that preserves the conormal bundle to σ (since both Kodaira- Spencer morphisms are the identity). Such an automorphism α is given by a section of F 1 (Ad q (P )). The eﬀect of such an automorphism α on the connection ∇ P is given log obtained by applying the morphism by adding to ∇ P the section of Ad c (P ) ⊗ O U ω U/S log Ad(∇ P ) : F 1 (Ad q (P )) → F 0 (Ad c (P )) ⊗ O U ω U/S (induced by the connection ∇ P ) to α. Thus, we obtain that the set of isomorphism classes of quadruples (π; σ; q i ; ∇ P ) that, relative to the bijection of Theorem 1.7, correspond to crystalline Schwarz structures with nilpotent monodromy are a torsor over the cokernel of Ad(∇ P ). On the other hand, by looking at the explicit representations of the subquotients of the ﬁltrations on Ad q (P ) and Ad c (P ) (given in the preceding paragraph), and using the fact that the Kodaira-Spencer morphism for ∇ P at σ is an isomorphism, we obtain that log Coker(Ad(∇ P )) ∼ = (ω U/S ) ⊗2 (−D) (We remark that here one uses the fact that p is odd, for when one computes Ad(∇ P ) from ∇ P , certain factors of 2 appear, and in order to get the above isomorphism, one needs for those factors of 2 to be invertible.) In other words, we have proven the following result: 38 Theorem 1.9. The étale sheaf of crystalline Schwarz structures on U log with nilpotent log ⊗2 ) (−D). monodromy is naturally a torsor over the sheaf (ω U/S Normalized P 1 -bundles with Connection Let us consider the P 1 -bundle π : P = P(J /J [3] ) → U , and section σ : U → P given by J /J [3] → J /J [2] , without any connection. Now, just as in the proof of Propo- sition 1.4, by taking residues, we obtain a natural duality between O P (−σ)/O P (−3σ) and π ∗ ω P/U (3σ) that respects the natural ﬁltrations on the two bundles. Let Q = P((J /J [3] ) ∨ ). As U -schemes, we may identify Q and P . Let O Q (1) denote the line bundle obtained from the deﬁnition of the projectivization; thus π ∗ O Q (1) = (J /J [3] ) ∨ . Let L = π ∗ {O Q (1)⊗ O Q τ P/U (−3σ)}. Then L is a line bundle on U and (J /J [3] ) ∨ = L⊗ O U π ∗ ω P/U (3σ). Thus, we obtain a natural isomorphism O P (−σ)/O P (−3σ) ∼ = (J /J [3] )⊗ O U L that respects ﬁltrations. If we then look at the quotients of both sides by their respective rank one subbundles (that make up the ﬁltrations), we obtain an isomorphism between log ∼ log ω U/S = (J /J [2] ) ⊗ L ∼ = ω U/S ⊗ L. That is, we get a natural trivial- = O P (−σ)/O P (−2σ) ∼ ization O U ∼ = L of L. In summary, we see that without any connection, we have constructed a natural ﬁltration-preserving isomorphism: γ : O P (−σ)/O P (−3σ) ∼ = J /J [3] Now let us suppose that we have a logarithmic connection ∇ P on π whose Kodaira- Spencer morphism at σ is an isomorphism. Then we get a commutative diagram like the one preceding Theorem 1.7. Pulling back by α R , then Ξ, and ﬁnally by σ L , we thus see that ∇ P induces an isomorphism: ζ(∇ P ) : O P (−σ)/O P (−3σ) ∼ = J /J [3] Now we saw above (Theorem 1.7) that ∇ P deﬁnes a Schwarz structure. But one “loose end” relative to the statement of Theorem 1.7 is that although Schwarz structures have no automorphisms, triples consisting of projective bundles with a section and a connection can have automorphisms. These automorphisms were the cause of the phenomenon (observed just before the statement of Theorem 1.9) that many diﬀerent ∇ P can give rise to the same Schwarz structure. Thus, it is convenient to have some sort of notion of a “normalized ∇ P ” such that each Schwarz structure arises from a unique normalized ∇ P . We choose the normalization as follows: Deﬁnition 1.10. We say that ∇ P is normalized if γ and ζ(∇ P ) are inverse to each other. Now as a formal consequence of this deﬁnition, we observe that we obtain the following normalized version of Theorem 1.7: 39 Theorem 1.11. If f : U → S is as above, then there is a natural one-to-one correspon- dence between crystalline Schwarz structures on U log and normalized logarithmic connec- tions on the P 1 -bundle π : P(J /J [3] ) → U whose associated Kodaira-Spencer morphism at the section σ : U → P (deﬁned by J /J [3] → J /J [2] ) is an isomorphism. Moreover, under this correspondence, the crystalline Schwarz structures with nilpotent monodromy correpond precisely to the triples such that ∇ P has nilpotent monodromy at the marked points. The Schwarzian Derivative Before proceeding, it is interesting to note that, as the name suggests, a crystalline Schwarz structure S allows one to deﬁne a Schwarzian derivative d S , as follows. Let × ω U log /S log ⊆ ω U log /S log denote the subsheaf consisting of sections that locally generate # ⊆ O U be the subsheaf consisting of functions φ such ω U log /S log as an O U -module. Let O U × that dφ is a section of ω U log /S log ⊆ ω U log /S log . Then our Schwarzian derivative will be a morphism of sheaves of sets: # ⊗2 d S : O U → ω U log /S log # Let θ be a section of O U over some étale V → U . Let us denote by j θ ∈ J /J [3] (V ) the 2- jet of θ (i.e., the Taylor expansion out to second order, modulo the constant term). By the # deﬁnition of O U , the image of j θ in J /J [2] (V ) is a local generator of the sheaf J /J [2] . By Proposition 1.6, j θ then deﬁnes a section s θ : V → P S . Taking the Kodaira-Spencer map of this section then deﬁnes an O V -linear morphism from τ U log /S log to the conormal bundle to s θ , which is simply ω U log /S log . This O V -linear morphism is thus given by multiplication ⊗2 by a section of ω U log /S log , which we take to be d S (θ). A simple calculation reveals that Proposition 1.12. If (as in Theorem 1.9) one modiﬁes the Schwarz structure S by log ⊗2 ) (−D)](U ) to obtain a Schwarz structure S , adding the square diﬀerential δ ∈ [(ω U/S then d S (θ) = d S (θ) + δ. We also have a biformal version of the Schwarzian derivative. Namely, we let O # bi be U the subsheaf of O U bi consisting of biformal functions φ that are of the form u + c, where u is a biformal uniformizer, and c is a biformal constant. Then we get a morphism of sheaves of sets: # ⊗2 ⊗2 L d bi S : O bi → (ω U log /S log ) = ω U log /S log ⊗ O U O U bi def U deﬁned as follows: If θ is a section of O # bi over some étale V → U , we let j θ be the U section of J /J [3] ⊗ O U O U bi which is the 2-jet of θ. Thus, j θ deﬁnes a section of s θ of P S L 40 ⊗2 L whose Kodaira-Spencer map is given by multiplication by a section of (ω U log /S log ) , which bi we take as d bi S (θ). Note that d S (i L (θ)) = i L (d S (θ)), and that if we modify the Schwarz structure by adding a square diﬀerential δ, then bi d bi S (θ) = d S (θ) + i L (δ) Remark. In the Appendix to this Chapter, we show that the deﬁnition just given for the Schwarzian derivative coincides with one-half the classical Schwarzian derivative, when U is the projective line. For the biformal version of the Schwarzian, we have an analogue of the classical result that the Schwarzian vanishes exactly on the formal functions that make up the projective structure of a Riemann surface. Indeed, let Ξ : P S L → P S R be the isomorphism deﬁned by the connection ∇ P S ; let α R : P S R → P S be the natural projection; and let def ζ : U bi = Spec(O U bi ) → P S be the morphism obtained by composing σ S L : U bi → P S L with Ξ and then α R . Since the deﬁnition of d S is functorial, d S applied to a function pulled back by ζ is ζ −1 of the “d S ” computed for projective bundles in the Appendix, i.e., one-half the classical Schwarzian. Thus, if θ is a section of S, then θ is the pull-back by ζ of a (degree ≤ 1) relative rational function for P S → U , so d bi S (θ) = 0. bi Conversely, suppose that d bi S (θ) = 0. Then the statement that d S (θ) = 0 means L that s θ is a horizontal section of P S . Thus it follows from the deﬁnition of a connection, together with the Poincaré Lemma in crystalline cohomology (see, e.g., [Kato], §6 for the log version) that Ξ(s θ ) is the pull-back via α R of a section t θ of P S . Now (after possible étale localization), we can ﬁnd a (degree ≤ 1) relative rational function φ for P S → U whose 2-jet at σ S is given by the section t θ . Since ζ maps the diagonal in U bi to σ S R , and def the formation of 2-jets is functorial, it thus follows that the 2-jet of ψ = ζ −1 (φ) deﬁnes a section s ψ of P S L which is equal to s θ when restricted to the diagonal U ⊆ U bi . But since both s ψ and s θ are horizontal, they must be equal. The biformal functions ψ and θ thus have 2-jets that deﬁne the same “line” in J /J [3] . Let z be a local coordinate on U . Let us denote by successive primes the derivatives of biformal functions (i.e., taken on the left) with respect to z. Then we obtain that ψ and θ are both invertible biformal functions such that ψ · θ = ψ · θ . It thus follows that (ψ /θ ) = 0, so θ = a ψ , where a is an invertible biformal constant. Thus, θ = a ψ + b, where b is a biformal constant. Since, by construction ψ ∈ S(V ), it follows that θ ∈ S(V ). Thus, we obtain the following “crystalline Schwarzian Poincaré Lemma:” Theorem 1.13. If θ is a biformal function, then d bi S (θ) = 0 if and only if θ is a section of S. 41 §2. Indigenous Bundles In this Section, we globalize the local considerations of §1, and are thus led to introduce “indigenous bundles” (as in [Gunning]). Let S log be a ﬁne log scheme, whose underlying scheme is connected noetherian. Let f log : X log → S log be proper, geometrically connected, and locally stable of dimension one. (Note that the ﬁrst two conditions are actually conditions on the underlying scheme morphism f .) We assume that the ﬁbers of f : X → S have arithmetic genus g ≥ 0, and exactly r ≥ 0 marked points (as in Deﬁnition 1.1 – note that these may only be deﬁned étale locally, however). Basic Deﬁnitions and Examples If 2g − 2 + r ≥ 1, then let M g,r be the moduli stack of stable curves of genus g, with r marked points, over Z, and let ζ : C → M g,r be the universal curve, with its r marked points s 1 , . . . , s r : M g,r → C. Note that M g,r has a natural log structure given log by the divisor at inﬁnity. Denote the resulting log stack M g,r . Also, by taking the divisor which is union of the s i and the pull-back of the divisor at inﬁnity of M g,r , we get a log structure on C; we call the resulting log stack C log . Also, ζ : C → M g,r extends naturally log to a morphism of log stacks ζ log : C log → M g,r . Deﬁnition 2.1. We shall say that f log : X log → S log is stable if there exists a classifying log morphism φ log : S log → M g,r such that X log ∼ = S log × M log C log . g,r Ultimately, we shall be concerned mainly with the case where f log is stable, but it is useful to realize that the deﬁnition, as well as many of the ﬁrst properties, of indigenous bundles can be made without these assumptions. Let π : P → X be a P 1 -bundle. If σ : X → P is a section, then we call the canonical height of σ the number 12 deg X/S (σ ∗ τ P/X ), where deg X/S denotes the relative degree over S of a line bundle on X, and τ P/X is the relative tangent bundle of π. If ∇ P is a logarithmic connection on P , then we call the morphism τ X log /S log → σ ∗ τ P/X given by diﬀerentiating σ by means of ∇ P the Kodaira-Spencer morphism at σ relative to ∇ P . Often, instead of dealing with P 1 -bundles with logarithmic connections, it will be more convenient to deal vector bundles: Thus, let E be a vector bundle equipped with a logarithmic connection ∇ E , whose rank is two and whose determinant is trivial. Then Theorem 1.7 motivates the following Deﬁnition 2.2. We shall say that (π : P → X, ∇ P ) is an indigenous bundle on X log if the monodromy at the marked points (which exist étale locally) is nilpotent, and there exists a section σ : X → P of π such that the Kodaira-Spencer morphism at σ with respect to ∇ P is an isomorphism. We shall say that (E, ∇ E ) is an indigenous vector bundle 42 on X log if the associated P 1 -bundle with logarithmic connection (P(E) → X, ∇ P(E) ) is an indigenous P 1 -bundle. We shall say that P → X (respectively, E) is intrinsic if there exists a logarithmic connection ∇ P (respectively, ∇ E ) on P → X (respectively, E) that makes (P → X, ∇ P ) (respectively, (E, ∇ E )) indigenous. We shall say that P → X (respectively, E) is locally intrinsic if it is intrinsic étale locally on S. Thus, in the vector bundle case, (E, ∇ E ) is indigenous if ∇ E has nilpotent monodromy at the marked points, and there exists a rank one subbundle F 0 (E) ⊆ E such that the Kodaira-Spencer morphism F 0 (E) → ω X log /S log ⊗ O X (E/F 0 (E)) (induced by ∇ E ) is an isomorphism. So far we have been discussing the hyperbolic case (2g − 2 + r ≥ 1); however, one can make the same deﬁnition for curves that are not hyperbolic. Example 1. Suppose that f : X → S is smooth, with no marked points, and that g = 0. Thus, f is a P 1 -bundle. Then the P 1 -bundle given by X × S X → X has a natural trivial connection, together with a natural section, the diagonal section. It is trivial to see that this triple satisﬁes the required properties for an indigenous bundle. Example 2. Suppose that f log : X log → S log has no marked points, and that its ﬁbers all have arithmetic genus one. Then consider the bundle E = ω X/S ⊕ O X (where ω X/S is the relative dualizing sheaf). Let L = f ∗ ω X/S . Thus, L is a line bundle on S, and f ∗ L ∼ = ω X/S . In particular, there exists on ω X/S a “trivial connection” ∇ ω obtained from tensoring the trivial connection on O X with f ∗ L. Let ∇ E be the connection on E which is the direct sum of ∇ ω and the trivial connection on O X . Let ∇ E be the connection on E given by adding to ∇ E the section of End(E) ⊗ ω X/S given by projecting E → ω X/S ∼ = (0, O X ) ⊗ ω X/S ⊆ E ⊗ ω X/S . Then one checks easily that if we take (P, ∇ P ) = P(E, ∇ E ), and σ : X → P to be given by (ω X/S , 0) ⊆ E, then we obtain an indigenous bundle on X log . Example 3. Let S log = Spec(Z) (with the trivial log structure); X log = M 1,1 (the moduli stack of one-pointed curves of genus one over Z), with its natural log structure. Let E be the vector bundle of rank two on X which is the ﬁrst de Rham cohomology module of the universal one-pointed curve of genus one. Then E has a natural logarithmic connection ∇ E , the so-called “Gauss-Manin connection.” There is also a natural Hodge def ﬁltration F 1 (E) ⊆ E, which deﬁnes a section σ : X → P = P(E). The pair (E, ∇ E ) forms the prototypical example of an indigenous bundle on X log . First Properties We now proceed to examine basic properties of such bundles. 43 Proposition 2.3. If f log : X log → S log and h log : Y log → S log are as stipulated at the beginning of the Section, and ζ log : Y log → X log is a log étale morphism of log schemes over S log that sends marked points to marked points, then the pull-back via ζ log of any indigenous bundle is again indigenous. Proof. This follows from the deﬁnitions. Proposition 2.4. If π : P → X is intrinsic, then the section σ : X → P is of canonical height 1 − g − 12 r. If X log → S log is a stable curve (so, in particular, 2g − 2 + r ≥ 1), then σ is the unique section of π of canonical height 1 − g − 12 r. We shall refer to σ as the Hodge section of π : P → X. Proof. The fact that the canonical height of σ is 1 − g − 12 r follows from the fact that the Kodaira-Spencer morphism is an isomorphism. Now suppose that X log → S log is stable. Let us ﬁrst assume that S is the spectrum of an algebraically closed ﬁeld. Suppose that σ : X → P also has canonical height 1−g − 12 r. Then it follows that its restriction to some irreducible component of X has negative canonical height. Since the restriction of σ ∗ τ P/X to any irreducible component has negative degree, it follows immediately from considering intersection numbers on P , together with the deﬁnition of “canonical height,” that there cannot exist two distinct sections of negative canonical height over that irreducible com- ponent. Thus, σ and σ must agree over that irreducible component. Now if there are any other irreducible components in X, then σ must have negative canonical height over some other irreducible component of X, in order for its canonical height over all of X to be 1 − g − 12 r. Thus, repeating this argument shows that σ = σ . Finally, let us observe that the space of deformations of σ is given by H 0 (X, σ ∗ τ P/X ), which is zero, since σ ∗ τ P/X has negative degree on every irreducible component of X. The result for general S then follows immediately from this by deformation theory. Now let us assume for the rest of the Section (unless stated otherwise) that there exists an odd prime p which is nilpotent on S, together with a PD-ideal I ⊆ O S . Proposition 2.5. If π : P → X is intrinsic, then P ∼ = P(J /J [3] ) (where J deﬁnes the diagonal in X bi ). Moreover, for any connection on π that makes it indigenous, the mon- odromy at a marked point s : S → X ﬁxes the section q s : S → P(J /J [3] ) of Proposition 1.6. Proof. This follows from Theorem 1.7 and Propositions 1.3 and 1.6. Note that the second statement uses the fact that p is odd. Proposition 2.6. Suppose that the number of marked point plus nodes on any geometric irreducible component of a ﬁber of X → S is even. Let (π : P → X, ∇ P ) be indigenous on 44 X log . Then étale locally on S, there exists an indigenous vector bundle (E, ∇ E ) on X log whose projectivization is (π : P → X, ∇ P ). Moreover, such an (E, ∇ E ) is unique up to tensor product with a line bundle with connection (L, ∇ L ) on X whose square is trivial. Proof. Consider the relative anticanonical bundle τ P/X on P . By Proposition 2.5, (after étale localization on S) there exists a line bundle G on P whose square is τ P/X . Now let us note that since the construction of the anticanonical bundle is canonical, it follows that the connection ∇ P on the P 1 -bundle induces a connection on the polarized P 1 -bundle (π : P → X, τ P/X ). Moreover, since the “moduli space” of line bundles G whose square is τ P/X is étale over X, it follows that the connection ∇ P on the P 1 -bundle π : P → X in fact induces a connection on the polarized P 1 -bundle (π : P → X, G). Thus, we def get a connection ∇ E on E = π ∗ G. Moreover, on P , we have a natural exact sequence 0 → ω P/X → (π ∗ E) ⊗ G −1 → O P → 0, which induces an isomorphism det(π ∗ E) ∼ = O P , hence an isomorphism det(E) ∼ = O X , which is easily seen to be horizontal. Finally, it is clear that the projectivization of (E, ∇ E ) is isomorphic to (π : P → X, ∇ P ). Now suppose that both (E, ∇ E ) and (E , ∇ E ) have the same projectivization (π : P → X, ∇ P ). Then E deﬁnes a line bundle G on P whose square is τ P/X and such that ∇ P induces a connection on the polarized P 1 -bundle (π : P → X, G). Similarly, E deﬁnes a line bundle G on P . Since we have horizontal isomorphisms G ⊗2 ∼ = τ P/X and (G ) ⊗2 ∼ = τ P/X , −1 it follows that if we let L = π ∗ ((G ) ⊗ G), then L gets a natural connection ∇ L such that the square of (L, ∇ L ) is trivial. Moreover, (E, ∇ E ) ∼ = (E , ∇ E ) ⊗ (L, ∇ L ). This completes the proof. In summary, the above Proposition tells us that (under the evenness assumption) up to étale localization on the base, it is essentially the same thing to give an indigenous P 1 - bundle or an indigenous vector bundle. Thus, in the future, we shall frequently simply speak of “indigenous bundles.” The same goes for intrinsic bundles. Existence and de Rham Cohomology The next step is to prove the existence of indigenous bundles, and to parametrize them. We begin with the proof of existence. For the rest of this Section, we shall assume that f log is stable. Thus, in particular, 2g − 2 + r ≥ 1. Theorem 2.7. For any r-pointed stable curve X log → S log of genus g, the P 1 -bundle P(J /J [3] ) is locally intrinsic. Proof. From Theorem 1.9, we know that the obstruction to the existence of a crystalline Schwarz structure with nilpotent monodromy on X log (locally on S) is given by a section log ⊗2 log ⊗2 of R 1 f ∗ (ω X/S ) (−D) over S. On the other hand, by Serre duality, R 1 f ∗ (ω X/S ) (−D) 45 is isomorphic to the dual of f ∗ τ X log /S log = 0, since the curve is stable. The Theorem now follows form Proposition 1.6 and Theorem 1.7. Next we wish to compute the de Rham cohomology of the P 1 -bundle with parabolic structure (π; q i ). Note that the exterior diﬀerential operator maps Ad(P ) (respectively, Ad q (P )) into Ad q (P ) (respectively, Ad c (P )). We deﬁne the parabolic de Rham cohomology (respectively, with compact supports) of Ad(P ) to be the hypercohomology of the complex log log (respectively, Ad q (P ) → Ad c (P ) ⊗ ω X/S ). Ad(P ) → Ad q (P ) ⊗ ω X/S Theorem 2.8. Let (P, ∇ P ) be an indigenous bundle on an r-pointed stable curve f log : X log → S log of genus g. Then the de Rham cohomology of Ad(P ) with its natural connec- tion (induced by ∇ P ) is as follows: (1) For cohomology without compact supports, we have (f DR ) ∗ (Ad(P )) = R 2 (f DR ) ∗ (Ad(P )) = 0; and we have a natural exact sequence log ⊗2 ) (−D) → R 1 (f DR ) ∗ (Ad(P )) → R 1 f ∗ τ X log /S log → 0 0 → f ∗ (ω X/S (2) For cohomology with compact supports, we have (for all i ≥ 0) a natural isomorphism R i (f DR ) c,∗ (Ad(P )) ∼ = R i (f DR ) ∗ (Ad(P )) In particular, (P ; ∇ P ) has no nontrivial automorphisms. Proof. To compute the de Rham cohomology, one uses the long exact cohomology se- quences induced by the ﬁltrations considered above, plus the fact that the Kodaira-Spencer morphism is an isomorphism. Now let α be an automorphism. Since (f DR ) ∗ (Ad(P )) = 0, all inﬁnitesimal automorphisms must vanish, so we may work over an algebraically closed ﬁeld. By passing to a tamely ramiﬁed covering of X ramiﬁed only at the marked points and nodes, we may assume that the hypotheses of Proposition 2.6 are satisﬁed. Then let L be a line bundle on P whose square is τ P/X . Since L ∼ = O P (σ) ⊗ O S M (for a line bundle M on S), and α always preserves σ, it follows that α preserves L. Thus, α arises from a horizontal section of End(π ∗ L) = O X ⊕ Ad(P ), hence is induced by multiplying π ∗ L by a section of O S . Thus, α is the identity, as desired. Finally, combining what we have done in this Section with Theorem 1.7, we obtain: Corollary 2.9. Let f log : X log → S log be an r-pointed stable curve of genus g. Then the set of crystalline Schwarz structures on X with nilpotent monodromy is in one-to-one 46 correspondence the set of isomorphism classes of indigenous P 1 -bundles on X log . More- over, the functor that assigns to T log → S log the set of crystalline Schwarz structures with log ⊗2 ) (−D). nilpotent monodromy on X T log = X log × S log T log is a torsor over f ∗ (ω X/S Indigenous Bundles of Restrictable Type Let f i log : X i log → S log (for i = 1, . . . , n) be an r i -pointed smooth curve of genus g i (where 2g i − 2 + r i ≥ 1 for all i). Suppose that we are given a graph Γ consisting of n vertices, numbered 1 through n. Let E i be the set of edges of the i th vertex. Suppose further that we are given an injection λ i : E i → {1, . . . , r i }. Then we can glue together the curves f i log : X i log → S log to form an r-pointed stable curve f log : X log → S log of genus g in such a way that the dual graph of f log is given by Γ, that is: (1) vertex i corresponds to f i log : X i log → S log , an irreducible component of X log ; (2) if is an edge running from vertex i to vertex j such that λ i ( ) = a and λ j ( ) = b, then corresponds to a node on X log obtained by gluing together X i log at the a th marked point to X j log at the b th marked point; (3) g and r can be computed combinatorially from Γ, the g i ’s, the r i ’s and the λ i ’s. : X i log → X log be the inclusion of X i log into X log as one of the irreducible Let μ log i components. Now let us suppose that we are given an indigenous bundle (π : P → X; ∇ P ) on X log . log ∗ Then it is not necessarily the case that (μ log i ) (π : P → X; ∇ P ) will be indigenous on X i . The problem is that since in general, marked points of X i log might be sent to nodes of X log (and not to marked points), there is no reason why the monodromy at such marked points of X i log should be nilpotent. We therefore make the following log ∗ Deﬁnition 2.10. If the (μ log for all i, then we i ) (π : P → X; ∇ P ) are indigenous on X i say that (π : P → X; ∇ P ) is of restrictable type. Now let us suppose that we are given indigenous bundles Π i = (π i : P i → X i ; ∇ P i ) on X i log . Note that for each marked point s : S → X i of an X i log , s ∗ P i has a canonical trivialization as a P 1 -bundle given by considering: (1) the Hodge section σ i : X i → P i (pulled-back by s); (2) the trivialization of s ∗ (σ i ∗ ω P i /X i ) given by the residue map; and (3) the section q s : S → s ∗ P i of Proposition 1.6. 47 It thus follows that we can glue together the Π i ’s by means of this canonical trivialization at the marked points to obtain an indigenous bundle Π = (π : P → X; ∇ P ) on X log . Moreover, by construction, Π is of restrictable type. Also, we can clearly reverse the procedure: Namely, if we start with an indigenous bundle Π on X log of restrictable type, we can reconstruct Π by restricting to the X i log ’s, and then regluing, in the fashion described in the preceding paragraph. Now let us deﬁne n (f i ) ∗ ω ⊗2 log def E = X i i=1 /S log (−D i ) where D i is the divisor of marked points on X i . Then we have the following result: Proposition 2.11. The étale sheaf on S of isomorphism classes of indigenous P 1 -bundles of restrictable type on X log is a torsor over the vector bundle E. Note that the rank of E is given by than 3g − 3 + r. n i=1 (3g i − 3 + r i ), which, in general, is strictly less §3. The Obstruction to Global Intrinsicity In §2, we saw that the P 1 -bundle P(J /J [3] ) is locally intrinsic on M g,r . In this Section, we study the obstruction (which, in general, is nonzero) to it being globally intrinsic over all of M g,r . The main point is a computation in Hodge cohomology which, in many respects, is similar to that of [Falt3], Lemma IV.4. Since, however, it is not literally the same as [Falt3], and certain technical aspects of the computation are diﬀerent, we provide a complete proof here. Introduction of Cohomology Classes We shall work over a ﬁeld K of characteristic zero, say Q p , until we state otherwise. Since we are only interested in certain intersection numbers, the base ﬁeld is essentially irrelevant. Let us consider the universal r-pointed stable curve of genus g, ζ : C → M g,r (over K). We would like to consider various cohomology classes on C and M g,r . The cohomology theory that we will use is Hodge theory, so all cohomology classes are to be understood as being Hodge-theoretic. Let π : P = P(J /J [3] ) → C be the P 1 -bundle which, as we saw in §2, is locally intrinsic. Let F = Ad(P ). Thus, F has a ﬁltration whose subquotients are given by: F 1 ∼ = L; F 0 /F 1 ∼ = O C ; F −1 /F 0 ∼ = L −1 48 where L = ω C log /M log . Let η = c 1 (L), the ﬁrst Chern class of L. Then we have: c 1 (F) = 0. g,r On the other hand, the second Chern class of F is given by: c 2 (F) = −η 2 2 Let r us compute ζ ∗ η . Let D i ⊆ C (where i = 1, . . . , r) be the marked points. Let D = i=1 D i . We shall write [D i ]; [D] for the respective cohomology classes on C. Let ξ = c 1 (ω C/M g,r ). Thus, η = ξ + [D]. Since diﬀerent D i ’s do not intersect, we have [D i ] · [D j ] = 0 if i = j. Also, by “taking the residue,” we see that ζ ∗ {(ξ + [D i ]) · [D i ]} = 0, r for all i. Thus, ζ ∗ {(ξ + [D]) · [D]} = 0. Let ψ i = ζ ∗ (ξ · [D i ]); ψ = i=1 ψ i ; θ = ζ ∗ ξ 2 . Then we obtain: ζ ∗ η 2 = ζ ∗ {(ξ + [D]) · ξ} = θ + ψ Now one knows from [AC] that for g ≥ 3, the restrictions of the classes θ and ψ to M g,r are linearly independent. We summarize this in a Lemma: Lemma 3.1. We have, on M g,r , ζ ∗ η 2 = θ + ψ. In particular, if g ≥ 3, then (ζ ∗ η 2 )| M g,r is nonzero. We shall see below that ζ ∗ c 2 (F) can be related to the obstruction to the existence of a global indigenous bundle on C. Thus, once we have done this, we will have proven that this obstruction is given by the relatively computable cohomology class −ζ ∗ η 2 on M g,r . Computation of the Second Chern Class Let us ﬁrst observe that P | D has a canonical trivialization as a P 1 -bundle given by using: (1) the Hodge section σ : C → P (restricted to D); (2) the trivialization of (σ ∗ ω P/C )| D given by the residue map; and (3) the section q s : D → P × C D of Proposition 1.6. Let us denote by C c log the log stack obtained by letting C be the underlying stack and log taking for the log structure the pull-back of the log structure of M g,r via ζ. Thus, we have an exact sequence on C: 0 → ζ ∗ Ω M log → Ω C c log → ω C/M g,r → 0 g,r 49 where the ﬁrst two sheaves of diﬀerentials are over K. In the future, we shall think of this exact sequence as deﬁning a one-step ﬁltration on Ω C c log . For i, j ≥ 0, let us deﬁne for any O C -module G: def H c i,j (C, G) = H i (C, G ⊗ O C ∧ j Ω C c log ) This cohomology is a sort of cohomology with compact supports outside D. Thus, by using the canonical trivialization of P | D referred to above, we see that the global obstruction to the existence of a logarithmic connection on π : P → C (for the log structure of C log ) which has normalized nilpotent monodromy at D deﬁnes a class κ ∈ H c 1,1 (C, Ad(P )). Now, by taking the trace of the square of κ, we get a class in tr(κ 2 ) ∈ H c 2,2 (C, O C ). If we then apply log def ζ ∗ , we get a class ζ ∗ tr(κ 2 ) ∈ H 1,1 (M g,r ) = H 1 (M g,r , Ω M log ). On the other hand, let us g,r denote by: S g,r → M g,r the Ω M log -torsor deﬁned by looking at the crystalline Schwarz structures with nilpotent g,r log monodromy on C (as in Corollary 2.9). This torsor thus deﬁnes a class Σ ∈ H 1,1 (M g,r ). The goal of this subsection is to prove the following: log Lemma 3.2. We have the following equality of classes in H 1,1 (M g,r ): Σ = 12 ζ ∗ tr(κ 2 ). Now let us note that H c 1,1 (C, Ad(P )) has two one-step ﬁltrations: one arising from the Leray-Serre spectral sequence applied to ζ, and the other arising from the ﬁltration deﬁned above on Ω C c log . Thinking in these terms, we see that we get a morphism: φ 00 : H c 1,1 (C, Ad(P )) → H 0 (M g,r , R 1 ζ ∗ (Ad(P ) ⊗ ω C/M g,r )) Now since we know that P → C admits a connection of the desired type on the ﬁbers of ζ, it follows that φ 00 (κ) = 0. Next let us consider the natural morphism: φ 10 : H c 1,1 (C, Ad(P )) → H 1 (C, Ad(P ) ⊗ ω C/M g,r ) Since φ 00 (κ) = 0, it follows from considering the Leray-Serre spectral sequence that φ 10 (κ) lies in H 1 (M g,r , ζ ∗ (Ad(P ) ⊗ ω C/M g,r )) → H 1 (C, Ad(P ) ⊗ ω C/M g,r ). In fact, by consid- ering only normalized connections (as in Deﬁnition 1.10), we can say more. Namely, it follows that φ 10 (κ) is actually the image under the morphism H 1 (M g,r , ζ ∗ (ω C log /M log ⊗ g,r ω C/M g,r )) → H (M g,r , ζ ∗ (Ad(P ) ⊗ ω C/M g,r )) (induced by ω C log /M log → Ad(P )) of the 1 g,r 50 class Σ (regarded as a class in H 1 (M g,r , ζ ∗ (ω C log /M log ⊗ ω C/M g,r )) by means of the tauto- g,r logical isomorphism Ω log ∼ )). log ⊗ ω = ζ ∗ (ω M g,r C/M g,r C log /M g,r Now let us consider the natural morphism: φ 01 : H c 1,1 (C, Ad(P )) → H 0 (M g,r , R 1 ζ ∗ (Ad(P ) ⊗ Ω C c log )) Since φ 00 (κ) = 0, it follows that the section φ 01 (κ) of R 1 ζ ∗ (Ad(P ) ⊗ Ω C c log ) lies in the image of the natural map ι : R 1 ζ ∗ (Ad(P ) ⊗ Ω M log ) → R 1 ζ ∗ (Ad(P ) ⊗ Ω C c log ) g,r In fact, we can say more. Since we are dealing with sheaves on M g,r , we can compute locally on M g,r . Let U → M g,r be étale. Let ∇ P be a logarithmic connection with normalized nilpotent monodromy (relative to ζ U : C U → U ) on P U = P × M g,r U . Then the obstruction to lifting ∇ P to a logarithmic connection relative to C U → Spec(K) is giving by subtracting the two pull backs of (P U → U ; ∇ P ) to the ﬁrst inﬁnitesimal neighborhood Δ U of the diagonal of U × K U . Note that it only makes sense to compare these two pull-backs because we have chosen a connection ∇ P , so that we can deal with crystals on Crys(C U /Δ U ) (where the structure morphism C U → Δ U is given by composing ζ U : C U → U with the diagonal embedding U → Δ U ). Thus, the diﬀerence between the pull-backs deﬁnes a section δ (over U ) of Ω M log | U ⊗R 1 (ζ U ) DR,∗ (Ad(P ), ∇ P ). Now if we compose the projection g,r R 1 (ζ U ) DR,∗ (Ad(P ), ∇ P ) → R 1 (ζ U ) ∗ τ C log /U log with δ, we get a morphism U (Ω M log ) ∨ | U = Θ M log | U → R 1 (ζ U ) ∗ τ C log /U log g,r U g,r Now it is a tautology that this morphism is none other than the isomorphism β derived log from deformation theory of the tangent space to M g,r with the ﬁrst cohomology group of the relative tangent bundle of ζ log . Thus, in summary, we have proven the following statement: (*) locally on M g,r , φ 10 (κ) is the image under ι of some local section ν of R 1 ζ ∗ (Ad(P ) ⊗ Ω M log ) whose image in R 1 ζ ∗ (τ C log /M g,r ) ⊗ Ω M log is g,r g,r the tautological isomorphism β. We are now ready to consider ζ ∗ tr(κ 2 ). We begin by using the observations of the preceding two paragraphs to compute what happens when we multiply various subquotients of the two ﬁltrations on H c 1,1 (C, Ad(P )) by each other: (1) If we multiply two elements in the image of H 1 (M g,r , ζ ∗ (Ad(P )) ⊗ Ω M log ), and take the trace, we get a (2, 2)-Hodge cohomology class on g,r 51 C which is the pull-back of such a class on M g,r ; thus, if we apply ζ ∗ to such a product, we get zero. (2) If we multiply an element in the image of H 1 (M g,r , ζ ∗ (Ad(P ))⊗Ω M log ) g,r by an element in the image of H 0 (M g,r , R 1 ζ ∗ (Ad(P ))⊗Ω M log ), and take g,r the trace, we get a class in H 1 (M g,r , R 1 ζ ∗ O C ⊗ ∧ 2 Ω M log ); since there g,r are no factors of ω C/M g,r in the wedge product, applying ζ ∗ again gives zero. (3) If we multiply two elements of H 0 (M g,r , R 1 ζ ∗ (Ad(P )) ⊗ Ω M log ), we g,r get zero since ζ has relative dimension one. (4) If we multiply an element in the image of H 1 (M g,r , ζ ∗ (Ad(P ))⊗Ω M log ) g,r by φ 10 (κ) ∈ H (M g,r , ζ ∗ (Ad(P ) ⊗ ω C/M g,r )), and take the trace, we get zero, since we are taking the trace of the product of a nilpotent section of Ad(P ) with a section of the same Borel subalgebra of Ad(P ). 1 (5) If we square φ 10 (κ) ∈ H 1 (M g,r , ζ ∗ (Ad(P )⊗ω C/M g,r )), we get zero since we are, in eﬀect, squaring nilpotent sections of Ad(P ). (6) If we multiply φ 10 (κ) ∈ H 1 (M g,r , ζ ∗ (Ad(P ) ⊗ ω C/M g,r )) by φ 01 (κ) ∈ H 0 (M g,r , R 1 ζ ∗ (Ad(P ) ⊗ Ω C c log )), then we are, in eﬀect, multiplying the class Σ by the tautological isomorphism β, so that we obtain Σ, regarded as a class in H 1 (M g,r , Ω M log ). g,r Thus, in summary, all of the possible contributions are zero, except for the last, which is 2Σ. This completes the proof of Lemma 3.2. On the other hand, by basic linear algebra, c 2 (Ad(P )) is −2 tr(κ 2 ), so we see that we have, in fact, proven the following: Lemma 3.3. We have ζ ∗ c 2 (Ad(P )) = −4Σ. Finally, putting this together with Lemma 3.1, we see that we have explicitly computed the class Σ in terms of well-known ﬁrst Chern classes of line bundles: Theorem 3.4. The torsor of Schwarz structures deﬁnes a class Σ ∈ H 1 (M g,r , Ω M log ) g,r which is equal to 14 (θ + ψ), where θ = ζ ∗ ξ 2 ; ψ = ζ ∗ (ξ · [D]); ξ = c 1 (ω C/M g,r ); and [D] is the cohomology class of the divisor of marked points. In particular, if g ≥ 3, then (in characteristic zero) S g,r → M g,r does not admit any sections, i.e., there are no canonical Schwarz structures on r-pointed smooth curves of genus g ≥ 3. Remark. Ideally, it would be nice to have an equality of classes not in H 1 (M g,r , Ω M log ), g,r but in some sort of cohomology with compact supports “H c 1 (M g,r , Ω M log )” (which should g,r 52 be isomorphic to H 1 (M g,r , Ω M g,r )). In order to do this, one would have to deﬁne some sort of appropriate sense in which Σ is compactly supported, i.e., one would have to deﬁne some sort of trivializations of S g,r → M g,r at inﬁnity. In fact, S g,r → M g,r does not (in general) have a canonical section over the divisor at inﬁnity. However, by considering indigenous bundles of restrictable type, one can show that, so to speak, “the more singular a curve gets, the more of a canonical trivialization one has for S g,r → M g,r .” For instance, if the curve is totally degenerate, i.e., it can be constructed by gluing together (as at the end of §2) a number of copies of P 1 with three marked points, then S g,r → M g,r does have a canonical trivialization, as follows immediately from Proposition 2.11 (since then the indigenous bundles of restrictable type form a torsor over the zero sheaf). Thus, in some sort of combinatorially complicated sense, by considering indigenous bundles of restrictable type, one can exhibit Σ as a cohomology class with compact supports. Unfortunately, however, the combinatorics involved get rather complicated in general, so we shall not carry this out explicitly, except in the case when the dimension of M g,r is one, where things are not so diﬃcult. The Case of Dimension One In this case, either g = r = 1 or g = 0, r = 4. Let D ∞ ⊆ M g,r be the divisor at inﬁnity. Since D ∞ is zero-dimensional, in this case we do have a canonical trivialization t ∞ of S g,r → M g,r over D ∞ . Now we shall give a new deﬁnition of cohomology with compact supports that takes into account this trivialization t ∞ . Let ω C log (respectively, ω (M log ) ) be g,r ! ! the subsheaf ω C/M g,r (−D ∞ ) (respectively, ω M log (−D ∞ )) of ω C/M g,r (respectively, ω M g,r ). g,r Let Ω C log be the inverse image of ω C log ⊆ ω C/M g,r via the morphism Ω C c log → ω C/M g,r . ! ! Thus, we have an exact sequence: 0 → ζ ∗ Ω M log → Ω C log → ω C log → 0 g,r ! ! which deﬁnes a ﬁltration on Ω C log . Then for i, j ≥ 0, let us deﬁne for any O C -module G: ! def H ! i,j (C, G) = H i (C, G ⊗ O C ∧ j Ω C log ) ! and for any O M g,r -module H: def H ! i,j (M g,r , H) = H i (M g,r , H ⊗ O M g,r ∧ j ω (M log ) ) g,r ! Note that the push forward map ζ ∗ naturally deﬁnes a morphism ζ ∗ : H ! 2,2 (C, O C ) → H ! 1,1 (M g,r , O M g,r ). Now the obstruction to putting a logarithmic connection on P → C with normalized nilpotent monodromy at the marked points and which is of restrictable 53 type at inﬁnity is given by a class κ ! ∈ H ! 1,1 (C, Ad(P )). On the other hand, the torsor S g,r → M g,r together with the trivialization t ∞ deﬁnes a class Σ ! ∈ H ! 1,1 (M g,r , O M g,r ). By the same proof as before, we have Σ ! = 12 ζ ∗ tr(κ 2! ). (In the present case, however, one might remark that in the six types of product considered previously, the ﬁrst two types (numbered (1) and (2)) of product vanish all the more trivially since they involve ∧ 2 ω M log , which is zero.) Also, just as before, we have c 2 (Ad(P )) = −2 tr(κ 2! ). Thus, we obtain: g,r Theorem 3.5. In H ! 1,1 (M g,r , O M g,r ) = K, we have Σ ! = 1 ζ ∗ η 2 4 where η = c 1 (ω C log /M log ). g,r Let us compute ζ ∗ η 2 in the case g = 1, r = 1. First we introduce the classes θ = ζ ∗ ξ 2 ; ψ = ζ ∗ (ξ · [D]); and λ = c 1 (ζ ∗ ω C/M g,r ). By Grothendieck-Riemann-Roch, θ = 0. On the other hand, sorting through the deﬁnitions, one sees that ψ = λ. Thus, we obtain that for g = 1, r = 1: Σ ! = log 1 λ 4 log log Next let N log : M 1,1 [2] → M 1,1 be the ﬁnite, log étale covering such that M 1,1 [2] is the moduli stack that parametrizes elliptic curves with level structure on the 2-torsion points. Let log log Λ log : M 1,1 [2] → M 0,4 be the log étale morphism given by sending an elliptic curve with a trivialization of its two torsion to the four-pointed curve of genus zero of which the elliptic curve is a double covering (with ramiﬁcation exactly at the four marked points). Moreover, Λ log admits a log section over any double covering of M 0,4 (since the obstruction to such a section lies in 2 (M 0,4 , Z/2Z)). Note that both N log and Λ log are, in fact, deﬁned over Z[ 12 ]. Also, H et note that (over Z[ 12 ]), we have an isomorphism Λ ∗ S 0,4 ∼ = N ∗ S 1,1 obtained by pulling-back and pushing forward indigenous bundles. We thus obtain the following result: Theorem 3.6. When M g,r is one-dimensional, the torsor S g,r → M g,r does not admit any section which is equal to t ∞ over D ∞ modulo any prime ≥ 5. 54 Proof. We have (Σ ! ) 1,1 = 14 λ 1,1 , and λ 1,1 = 12 c 1 (ω M log ), so (Σ ! ) 1,1 = 18 c 1 (ω M log ). Thus, 1,1 ∗ log 1,1 ∗ is log étale, and N (Σ ! ) 1,1 is an invertible multiple of Λ (Σ ! ) 0,4 , it suﬃces to since N show that: (1) in Hodge cohomology modulo p (for a prime p ≥ 5), Λ ∗ c 1 (ω M log ) = 0,4 N ∗ c 1 (ω M log ) is nonzero; in fact, since Λ log admits a section over any 1,1 log double covering of M 0,4 , it suﬃces merely to show that c 1 (ω M log ) is 0,4 nonzero modulo p; (2) the formation of H ! 1,1 (M 0,4 , O M 0,4 ) commutes with base change modulo p (for p an odd prime). But both (1) and (2) follow immediately from the fact that M 0,4 is just P 1 , with D ∞ = {0, 1, ∞}. (Note that there is a slight subtlety here in that (2) is not immediately obvious for M 1,1 since it is a stack; this is why we choose to verify the assertions of the Theorem by means of M 0,4 .) Appendix: Relation to the Complex Analytic Case In this Appendix, we make the connection between the theory of Schwarz structures discussed here and the classical notion of the Schwarzian derivative in complex analysis. Let K be an algebraically closed ﬁeld of characteristic zero (such as, for instance, the complex numbers C). Let X → Spec(K) be a smooth, proper, connected curve. Let P = P(J /J [3] ) be the usual P 1 -bundle on X, and let ∇ P be a connection on P → X that makes it indigenous. Then just as in §1, we can form the Schwarzian derivative: # 2 D : O X → ω X/K Then the purpose of this Appendix is to show that when X = P 1 ; z is the standard rational 2 is trivialized by (dz) 2 , then D(φ) is given (up to a factor of function on P 1 ; and ω X/K two) by the classical formula for the Schwarzian derivative. First, let us note that when X = P 1 , there exists only one connection ∇ P on π : P → X that makes it indigenous. Indeed, this follows from the fact that Ad(P ) ⊗ O X ω X/K has no sections for degree reasons, plus the fact that the extension 0 → J [2] /J [3] → J /J [3] → J /J [2] → 0 does not split (since the extension class is the Hodge-theoretic ﬁrst Chern class c 1 (ω X/K ), which is nonzero). It thus follows that (P ; ∇ P ) is necessarily isomorphic to the indigenous 55 bundle constructed in Example 1 of §2. Thus, P ∼ = X × K X (where we regard the projection to the second factor as the structure morphism to X); let us ﬁx such an isomorphism for the rest of the discussion. Also, the Kodaira-Spencer morphism at the diagonal section σ Δ : X → X × K X ∼ = P is an isomorphism. Let z be the standard rational function on X. Let U ⊆ X be the complement of inﬁnity. Thus, z is regular on U . We shall work mainly on U . For i = 1, 2, let p i : P ∼ = X × K X → X be the projection to the i th factor. Let ζ be the relative rational function on P | U → U given by p ∗ 1 (z) − p ∗ 2 (z). Let us d . Then clearly, simply denote by ∇ the result of applying ∇ P in the tangent direction dz 2 ∇(ζ) = −1. Let η = 1/ζ. Thus, ∇(η) = η . Now if we regard P as P(J /J [3] ), and s is a section given by [a dz, b (dz) 2 ] (where a, b ∈ O X (V ), for some open V ⊆ U ), then we have ζ(s) = a/b. Indeed, both sides of this equation deﬁne relative rational functions on P | U → U . The right-hand side has a simple pole at the section [dz, 0 · (dz) 2 ], which, by computing residues as in the deﬁnition of the isomorphism of Proposition 1.4, corresponds to the section ∞ × U . Thus, both sides of the equation have a simple pole at ∞ × U , and the same 1-jet at σ Δ , hence are equal. In particular, η(s) = b/a. # (V ). The 2-jet j φ of φ is given by φ dz + 12 φ (dz) 2 . Now suppose we are given φ ∈ O X φ Thus, if s φ is the section of P over V that is deﬁned by j φ , we have η(s φ ) = 2φ . Thus, we compute: ( φ ) = ∇(η(s φ )) 2φ = {∇(η)}(s φ ) + η(∇(s φ )) = η 2 (s θ ) + D(φ) (dz) −2 = (φ ) 2 + D(φ) (dz) −2 2 4(φ ) Expanding the derivative on the left and rearranging terms, we get: φ (φ ) 2 (φ ) 2 − − } (dz) 2 2φ 2(φ ) 2 4(φ ) 2 1 φ 3(φ ) 2 = { − } (dz) 2 2 φ 2(φ ) 2 D(φ) = { We have thus shown the following result: Theorem A. On P 1 , the Schwarzian derivative deﬁned at the end of §1 is equal to one-half the classical Schwarzian derivative. For a treatment of the classical Schwarzian derivative, we refer to [Lehto]. 56 Chapter II: Indigenous Bundles in Characteristic p §0. Introduction In this Chapter, we study indigenous bundles in characteristic p. In particular, we will be concerned with how these bundles interact with Frobenius. Our main tool for studying this interaction will be the p-curvature. We begin in §1 by studying FL-bundles, which are a special kind of rank two vector bundle with connection on a curve that corresponds to a lifting of the curve modulo p 2 . In §2, we deﬁne the Verschiebung map on indigenous bundles to be the determinant of the p-curvature of the indigenous bundle. It turns out that (essentially) indigenous bundles arise from FL-bundles precisely when their Verschiebung vanishes. Since it is precisely this sort of indigenous bundle – which (following [Katz]) we call nilpotent – that corresponds to an MF ∇ -object in the sense of [Falt], it is worthwhile deﬁning and studying the moduli space N g,r of such bundles. In order to study N g,r , we make two fundamental calculations (Theorems 2.3 and 2.13) concerning the Verschiebung. The ﬁrst tells us that the Verschiebung is ﬁnite and ﬂat, of degree p 3g−3+r , and the second calculates the derivative the Verschiebung in terms of invariants of the indigenous bundle which are easier to compute. In §3, we deﬁne the hyperbolic (higher genus) analogue of an ordinary elliptic curve: namely, we say that a hyperbolic curve is hyperbolically ordinary if it admits a nilpotent indigenous bundle at which the derivative of the Verschiebung is an isomorphism. Using the general machinery developed in §2, we then do a number of computations involving totally degenerate curves and elliptic curves which reveal that: (1) the hyperbolically ordinary locus of M g,r is open and dense (Corollary 3.8); (2) if one applies the deﬁnition of ordinariness in terms of indigenous bun- dles to the case of elliptic curves, one recovers the classical deﬁnition of an ordinary elliptic curve (Theorem 3.11); and (3) (at least if g ≥ 3, and p is suﬃciently large then) each irreducible component of N g,r that passes through a certain canonical nilpotent indigenous bundle on a totally degenerate curve has degree ≥ 2 over M g,r ; thus, there is no canonical choice of a nilpotent indigenous bundle on a generic r-pointed stable curve of genus g (Proposition 3.13). We end the Chapter with the observation that (3) is interesting in the sense that it con- stitutes a deviation from the behavior that one might expect by analogy to the complex case. 57 §1. FL-Bundles In this Section, we develop the theory of a certain kind of rank two bundle, which we call an FL-bundle, which arises from looking at the Cartier isomorphism of a curve. It turns out that the space of such bundles can also be used to parametrize the inﬁnitesimal deformations of a curve to Z/p 2 Z. The material we present here is essentially “well-known” (see, e.g., [Kato], §4), although our point of view is a little diﬀerent. Let p be a prime. Let S be a noetherian scheme over F p . Let us assume that we are given a ﬁne log structure on S, and let us denote the resulting log scheme by S log . Let us denote the absolute Frobenius ([Kato], §4) of S log by Φ S log : S log → S log . Let f log : X log → S log be an r-pointed stable curve of genus g (as in Chapter I, Deﬁnition 2.1, so 2g − 2 + r ≥ 1). In general, we shall denote by means of a superscript “F ” the result of base-changing by Φ S log . Let def Φ X log /S log : X log → (X log ) F = X log × S log ,Φ S log S log be the relative Frobenius. Deformations and FL-Bundles We begin by reviewing the Cartier isomorphism (as in [Kato], Theorem 4.12). Since a curve is one-dimensional, this amounts to the existence of an exact sequence of sheaves on X: log log F → (Φ X log /S log ) −1 (ω X/S ) → 0 0 → (Φ X log /S log ) −1 O X F → O X → ω X/S def where the morphism in the middle is the exterior diﬀerentiation operator d. Let Q = log . Then note that since the above exact sequence is functorial with to base- d(O X ) ⊆ ω X/S log → S log , the formation of Q is likewise functorial with respect to base-change. change T We would like to consider what happens to this exact sequence when it is tensored over (Φ X log /S log ) −1 O X F with (Φ X log /S log ) −1 (τ X log /S log ) F . Let T = (Φ X log /S log ) ∗ (τ X log /S log ) F . We then obtain (by using the long exact cohomology sequence for higher direct images) the following two exact sequences of sheaves on S: log 0 → O S → R 1 f ∗ Q ⊗ O X T → R 1 f ∗ ω X/S ⊗ O X T 0 → R 1 f ∗ (τ X log /S log ) F → R 1 f ∗ T → R 1 f ∗ Q ⊗ O X T → 0 where we use the fact that f ∗ (Q ⊗ Φ −1 X log /S log O X F log F τ X log /S log ) → f ∗ (ω X/S ⊗ O X T ) = 0 by degree considerations. Now let us note that T has a natural logarithmic connec- tion ∇ T obtained by declaring the sections of the subsheaf (Φ X log /S log ) −1 (τ X log /S log ) F ⊆ (Φ X log /S log ) ∗ (τ X log /S log ) F = T to be horizontal. Thus, by using the above exact sequences, we can compute the ﬁrst de Rham cohomology module of T (where we always understand T to be equipped with the connection ∇ T ). 58 Proposition 1.1. We have an exact sequence: 0 → R 1 f ∗ (τ X log /S log ) F → R 1 f DR,∗ (T ) → O S → 0 which is functorial with respect to base-change T log → S log . In particular, R 1 f DR,∗ (T ) is a vector bundle of rank 3g − 2 + r on S. Finally, R 1 f DR,∗ (T ) ⊆ R 1 f ∗ (T ). Let us denote by A the R 1 f ∗ (τ X log /S log ) F -torsor on S deﬁned by the above exact sequence. Let S log be a ﬁne log scheme whose underlying scheme is ﬂat over Z/p 2 Z, log → S log be an r-pointed stable curve of and such that S log ⊗ Z/pZ = S log . Let X genus g lifting f log . Then for any r-pointed stable curve Y log → S log of genus g that lifts (X log ) F → S log , we can associate a section θ Y of A as follows. Consider the obstruction log → Y log . to lifting the relative Frobenius Φ X log /S log : X log → (X log ) F to a morphism X This deﬁnes a section θ Y of R 1 f ∗ (T ). Observe that θ Y is independent of the choice of lifting log . Indeed, this follows from the fact that locally, if one changes the lifting X log , the X log obstruction cocycle will change by a derivation of X applied to a function pulled-back log F via Φ X log /S log from (X ) . But this will always give zero. This proves that θ Y depends only on Y log . Let us also observe that θ Y actually deﬁnes a section of R 1 f DR,∗ (T ) ⊆ R 1 f ∗ (T ). Indeed, to see this, we reason as follows. We work with bianalytic functions, as in Chapter I, §1. Then the inverse image via the relative Frobenius of O (X bi ) F in O X bi coincides with both i L {(Φ X log /S log ) −1 O X F } and i R {(Φ X log /S log ) −1 O X F }. Thus, the two pull-backs (from the right and left) of θ Y to O X bi both correspond to the obstruction to lifting i L {(Φ X log /S log ) −1 O X F } = i R {(Φ X log /S log ) −1 O X F } to a O S -ﬂat subalgebra (with log structure) of O X bi . This shows that θ Y is horizontal. log → S log of (X log ) F → S log . Then Now suppose that we consider another lifting Z the diﬀerence between the liftings Y log and Z log naturally deﬁnes a section θ Y Z of the vector bundle R 1 f ∗ (τ X log /S log ) F . It follows immediately from the deﬁnition of an obstruc- tion class that θ Y = θ Z + θ Y Z . Thus, if there existed a lifting Y log such that θ Y is a section of the subsheaf R 1 f ∗ (τ X log /S log ) F ⊆ R 1 f DR,∗ (T ), then it would follow that there exists another lifting Z log such that θ Z = 0. Thus, the relative Frobenius would lift to a log log morphism Ψ : X → Z . But then by pull-back, Ψ would induce a nonzero morphism log F log of (Φ X log /S log ) ∗ (ω X/S ) into ω X/S , which, by degree considerations, is absurd. Thus, we 1 conclude that no θ Y lies in R f ∗ (τ X log /S log ) F ⊆ R 1 f DR,∗ (T ). In other words, for every lifting Y log , θ Y deﬁnes a section of A. Let D be the R 1 f ∗ (τ X log /S log ) F -torsor over S of liftings of (X log ) F → S log to S log . Then, we see that we have deﬁned a canonical morphism of R 1 f ∗ (τ X log /S log ) F -torsors F : D→A Since any morphism of torsors is necessarily an isomorphism, we see that we have proven the following result: ([Kato], Theorem 4.12 (2)) 59 Proposition 1.2. The canonical morphism F : D → A is an isomorphism. Let (E, ∇ E ) be a rank two vector bundle on X with a connection ∇ E (relative to f log ). Suppose that there exists a horizontal exact sequence 0 → T → E → O X → 0 Then this exact sequence deﬁnes a section η of R 1 f DR,∗ (T ) over S. Deﬁnition 1.3. We shall call (E, ∇ E ) an FL-(vector) bundle if η maps to O S × ⊆ O S under the map R 1 f DR,∗ (T ) → O S of Proposition 1.1. We shall call a P 1 -bundle with connection (P, ∇ P ) on X log an FL-(P 1 )-bundle if it can be written étale locally on S as the P 1 -bundle associated to an FL-vector bundle. Remark. The letters “FL” stand for Frobenius lifting. Since a FL-bundle deﬁnes a section of the torsor A, it follows by Proposition 1.2 that it also deﬁnes a lifting Y log → S log of (X log ) F → S log . Also, we shall see below (Corollary 1.5) that, at least if S is reduced, then if a horizontal exact sequence as above exists, it is necessarily unique. The p-Curvature of an FL-Bundle Let us assume for the rest of the Section that p is odd. Throughout this Chapter the notion of the p-curvature of a bundle with connection in characteristic p will play an important role. We refer to [Katz], §5, 6; [Ogus], §1.2, 1.3 for basic facts concerning the p-curvature. ([Katz], of course, does not handle the arbitrary “log-smooth” case, but the deﬁnitions and proofs (of the properties that we will need) go through without change. At any rate, on the sorts of curves that we are working with, the theory of [Katz], §5, 6, is literally valid on an open, schematically dense subset, and many assertions can be checked after restriction to such an open subset.) Let (E, ∇ E ) be an FL-bundle. We would like to compute the p-curvature of (E, ∇ E ). The p-curvature P will be a horizontal section of T ∨ ⊗ O X Ad(E). Occasionally, we shall think of P as a morphism Ad(E) → T ∨ or a morphism T → Ad(E) (using the fact that Ad(E) is self-dual). By abuse of notation, we shall also refer to these morphisms by the notation P. Now, ﬁrst of all, since ∇ E stabilizes the ﬁltration T ⊆ E, by functoriality, P also respects this ﬁltration. Secondly, since T and O X clearly have p-curvature zero, P not only respects the ﬁltration, but is nilpotent, i.e., P : T → Ad(E) maps into the subbundle T ⊆ Ad(E) (given by endomorphisms of E obtained by projecting E → O X , mapping O X to T , and then injecting T → E). Thus, P basically amounts to a morphism from T to T , i.e., a section of f ∗ O X = O S . Proposition 1.4. Assume that p is odd. If (E, ∇ E ) is an FL-bundle on X log , then P : T → Ad(E) is given by multiplication T → T by −1, followed by the inclusion T → Ad(E). 60 Proof. Since the universal example of an FL-bundle on an r-pointed stable curve of genus g is given by the torsor A over M g,r , which is smooth, it suﬃces to check the assertion after restriction to a closed point of this universal A. Thus, we can assume that S = Spec(k), where k is a ﬁnite ﬁeld. Then S has a canonical lifting to a ﬂat scheme over Z/p 2 Z, namely, S = Spec(W (k)/p 2 W (k)) (where W (k) is the ring of Witt vectors with coeﬃcients in k), → S, we may and S has a natural Frobenius lifting Φ S . Thus, for any smooth scheme U , that is, a S-linear → U F speak of a Frobenius lifting (mod p 2 ) on U morphism Φ : U def whose reduction modulo p is the relative Frobenius U → U F of U = U ⊗ Z/pZ. Now let us take U to be an aﬃne open subset U ⊆ X, at which f is smooth, and → S be a smooth lifting of U , and let t be a which contains no marked points. Let U local coordinate on U . By the interpretation of FL-bundles in terms of obstructions to Frobenius liftings, we may compute E by using as follows: Over U , E| U ∼ = T | U ⊕ O U . Let us write sections of E relative to the decomposition T | U ⊕ O U and the basis given by (Φ −1 ( d ) F , 0); (0, 1) ; and let us denote by ∇ the connection ∇ E applied in the X log /S log dt d . Then ∇ is given by adding to the direct sum connection the matrix direction dt 0 1 p Φ 0 0 where the map Φ is the derivative (with respect to t) of some local Frobenius liﬁng . Since t F → (1 + t) p − 1 is a Frobenius lifting, Φ(t F ) must be of the form Φ on U (1+t) p −1+f (t), for some function f (t) on U . Therefore, p 1 Φ is of the form (1+t) p−1 +f (t). This gives ∇(1, 0) = 0; ∇(0, 1) = ((1 + t) p−1 + f , 0). Therefore, (∇) p (1, 0) = 0 and d p ∇ p (0, 1) = (p − 1)!(1, 0) (since ( dt ) f = 0). Finally, it follows easily from using the fact × that F p is a cyclic group that (p − 1)! = −1 (in F p ). This completes the proof. Corollary 1.5. Assume that p is odd. Let (E, ∇ E ) be a rank two vector bundle with logarithmic connection on X log (over S log ) deﬁned by a section η of R 1 f DR,∗ (T ). Then (E, ∇ E ) is an FL-bundle if and only if its p-curvature is nonzero at some point of every ﬁber of f : X → S. Proof. The “only if” part follows from Proposition 1.4. On the other hand, suppose that the image of η under the map R 1 f DR,∗ (T ) → O S of Proposition 1.1 vanishes at a point. By restricting to that point, we may assume that S is the spectrum of a ﬁeld, and that the image of η in O S vanishes identically. But then, it follows from the exact sequence of Proposition 1.1 that (E, ∇ E ) is the pull-back of a bundle under Frobenius. Then its p-curvature must vanish identically, which contradicts our assumption. Corollary 1.6. Assume that p is odd and that S is reduced. Let (E, ∇ E ) be an FL-bundle. Let U ⊆ X be an open subset; (L, ∇ L ) be a line bundle with logarithmic connection on U log ; 61 and ι : L → E| U be a horizontal morphism of O U -modules with logarithmic connections. Then ι factors through the injection T | U → E| U in the deﬁnition of E as an FL-bundle. Proof. By shrinking U , we may assume that f : X → S is smooth on U , and that U stays away from the marked points. We may also assume that the composite of ι with the projection E| U → O U is an isomorphism. Thus, we obtain a horizontal isomorphism of line bundles L → O U . But this implies that (L, ∇ L ) has p-curvature zero. Thus, we get a horizontal isomorphism L ⊕ T | U → E| U . Since the left-hand side has p-curvature zero, the same is true of the right-hand side. But this contradicts Proposition 1.4. Remark. It is not diﬃcult to construct counterexamples to Corollary 1.6 if one does not assume that S is reduced. §2. The Verschiebung on Indigenous Bundles In this Section, we deﬁne a “Verschiebung” morphism on the space S g,r of Schwarz structures that takes values in the space of square diﬀerentials (twisted by Frobenius). We then prove various basic properties of this morphism, such as computing its derivative. This computation reveals that the derivative looks rather like the Verschiebung morphism for the Jacobian of the curve, thus justifying the terminology. On the other hand, as we shall see in §3, unlike the Verschiebung of the Jacobian which only pertains to H 1 of the curve, the Verschiebung on indigenous bundles pertains to a nonabelian invariant of the curve, namely, the nilpotent indigenous bundles on the curve. It turns out that the study of nilpotent indigenous bundles, and thus of the Verschiebung on indigenous bundles are central to understanding uniformization theory in the p-adic context. In this Section, M g,r (respectively, S g,r ) will denote the moduli stack of r-pointed curves of genus g (respectively, equipped with a Schwarz structure) over F p . We assume throughout this Section that p is odd. The Deﬁnition of the Verschiebung Let S log be a ﬁne noetherian log scheme over F p . Let f log : X log → S log be an r- pointed stable curve of genus g. Let D ⊆ X be the divisor of marked points. Let (E, ∇ E ) be an indigenous bundle on X log (see §2 of Chapter I for more on such bundles). We remark here that throughout this paper, when we do various manipulations with indigenous bundles, it will be simpler to work with vector bundles, rather than P 1 -bundles. Of course, indigenous vector bundles only exist under certain conditions (cf. Proposition 2.6 of Chap- ter I), but this will not pose any problem, since we can always either Zariski localize on the curve, or pass to some sort of covering of the curve, and then descend for the ﬁnal result. Thus, in the future, for the rest of the paper, we shall act as if this problem does not exist, and always work with indigenous vector bundles, when it is technically more comfortable to do so. 62 We maintain the notation of §1 for the various Frobenius morphisms and for T = (Φ X log /S log ) ∗ (τ X log /S log ) F . Let P E : T → Ad(E) be the p-curvature of (E, ∇ E ). Consider the composite of P E with its dual P E ∨ . This composite is a horizontal morphism T → T ∨ , log ⊗2 F hence deﬁnes a section of (f ∗ (ω X/S ) ) . Let V E be − 12 times this section. Another way to put the deﬁnition of V E is as follows: We consider the square (P E ) 2 : (T ) ⊗2 → End(E) of P E , take the trace, and multiply by − 12 . Yet another way to put the deﬁnition of V E is that it is the determinant of P E (regarded as a map (T ) ⊗2 → O X ). Proposition 2.1. Assume that S is reduced. Then V E is zero if and only if the image of P E consists of nilpotent endomorphisms of E. Proof. Immediate from the deﬁnitions. Thus, we may think of V E as a measure of how nilpotent P E is. Note, in particular, that at a marked point, by deﬁnition ∇ E has nilpotent monodromy, so the p-curvature is already nilpotent there. Thus, V E has zeroes at all the marked points. By abuse of notation, we log ⊗2 ) (−D)) F . shall denote by V E the resulting section of (f ∗ (ω X/S log ⊗2 Deﬁnition 2.2. We shall refer to the section V E of (f ∗ (ω X/S ) (−D)) F as the Ver- schiebung of the indigenous bundle (E, ∇ E ). log Thus, in the universal case, when f log : C log → M g,r is the universal r-pointed stable curve of genus g, we obtain a morphism of M g,r -schemes: V g,r : S g,r → Q g,r where Q g,r is the geometric vector bundle corresponding to (i.e., Spec of the symmetric log ⊗2 ) (−D)) F . Note that both S g,r and Q g,r are of dimension algebra of the dual of) (f ∗ (ω X/S 3g−3+r over M g,r . The rest of this Chapter will be devoted to studying this Verschiebung morphism V g,r . Let us begin with some observations concerning the degree of V g,r as a polynomial map. Let ∇ E = ∇ E + θ be a logarithmic connection on E that makes it an indigenous bundle; here we may assume that θ is an Ad(E)-valued diﬀerential which deﬁnes a square- nilpotent endomorphism of E and which corresponds to a square diﬀerential θ SD . Thus, θ 2 = 0. If we then compute the p-curvature of (E, ∇ E ) by, say, working locally on U ⊆ X where there is a local coordinate x, and letting ∇ (respectively, ∇ ; θ x ) denote ∇ E def d (respectively, ∇ E ; θ) applied in the direction ∂ = dx , we note that because θ x 2 = 0, all the terms that involve θ x more than 12 (p + 1) times must vanish. Moreover, there is only one term that involves θ x exactly 12 (p + 1) times, namely: 63 θ x ∇ θ x ∇ . . . ∇ θ x that is, alternating θ x ’s and ∇’s, with a total of 12 (p + 1) copies of θ x and 12 (p − 1) copies of ∇. For future reference let us call this term ξ. Note that since θ x 2 = 0, any string θ x ∇ θ x can be rewritten θ x [∇, θ x ] (where the brackets denote the commutator), and that this commutator [∇, θ x ] is a linear operator (that is, linear over O X ). Moreover, this linear operator [∇, θ x ] preserves the Hodge ﬁltration of E. Thus, ξ may be rewritten as θ x times [∇, θ x ] to some power. Since [∇, θ x ] preserves the Hodge ﬁltration, it thus follows that ξ is a linear operator on E which is nilpotent with respect to the Hodge ﬁltration. In particular, ξ 2 = 0. At any rate, we may at least conclude that in the expression for the trace of the square of the p-curvature of ∇ E , θ occurs no more than p times in each term. We thus obtain the following result: (*) Relative to the aﬃne structures of S g,r and Q g,r , the pull-backs of the aﬃne variables on Q g,r via the morphism V g,r are polynomials in the aﬃne variables of S g,r of degree ≤ p. In fact, we would like to conclude a stronger result, namely that the degree is exactly p. In order to do this, we need to enlist the aid of Jacobson’s formula (see, e.g., [Jac], pp. 186-187): This formula states that if a and b are elements of an associative ring R of characteristic p, then p p p (a + b) = a + b + p−1 s i (a, b) i=1 where the s i (a, b) are given by the formula: p−1 (ad(ta + b)) (a) = p−1 is i (a, b)t i−1 i=1 computed in the ring R[t], where t is an indeterminate. In our case, we let b = ∇ and a = θ x , and we wish to compute the s j (a, b), where j = 12 (p − 1). Let η be the coeﬃcient multiplying t j−1 in the expression (ad(ta + b)) p−1 (a). Let α = ad(a); β = ad(b). Then the terms in η look like 12 (p − 3) copies of α and 12 (p + 1) copies of β applied to a in some order. Now ultimately, in order to compute the Verschiebung, we are interested in computing tr(η · ξ). Let τ be one of the terms that make up η. Now we separate the analysis of τ into two cases: the case where τ begins with an α, and the case where τ begins with a β. Suppose that τ begins with an α. Thus, τ looks like α γ 1 . . . γ p−2 a 64 where each γ i is either α or β. Now let us note that when one applies α = ad(θ x ) or β = ad(∇) to a linear operator on E, one gets back a linear operator. Moreover, α applied to any linear operator on E yields a linear operator that preserves the Hodge ﬁltration on E. Thus, we conclude that τ is a linear operator on E that preserves the Hodge ﬁltration. Since ξ is nilpotent with respect to the Hodge ﬁltration, it thus follows that tr(τ · ξ) = 0. Now suppose that τ begins with a β: β γ 1 . . . γ p−2 a Let us denote by σ the linear operator on E given by γ 1 . . . γ p−2 a (i.e., we leave oﬀ the initial β). We would like to show that σ preserves the Hodge ﬁltration on E. To see this, ﬁrst note that among the γ i ’s, the number of β’s is exactly one greater than the number of α’s. Also, note that (by Griﬃths transversality) relative to the Hodge ﬁltration on Ad(E), applying β decreases the ﬁltration index l (in F l (Ad(E))) by at most one, while applying α always increases the ﬁltration index l by one. Thus, since a = θ x ∈ F 1 (Ad(E)), it follows that σ ∈ F 0 (Ad(E)), i.e., it ﬁxes the Hodge ﬁltration on E, and so σ · ξ is nilpotent with respect to the Hodge ﬁltration. In particular, tr(σ · ξ) = 0. Since the trace map is horizontal, we thus obtain that tr([∇, σ · ξ]) = 0. Therefore, when we multiply τ = β(σ) by ξ and take the trace, we get tr(τ · ξ) = tr([∇, σ · ξ]) − tr(σ · [∇, ξ]) = −tr(σ · [∇, ξ]) In other words, tr(τ · ξ) depends only on the images of σ and [∇, ξ] in F 0 /F 1 (Ad(E)). To compute the image of σ in F 0 /F 1 (Ad(E)), we must analyze σ in greater detail. Now we saw that when we compute σ by applying α’s and β’s to a, β decreases the ﬁltration index by at most one. The only time it fails to decrease the ﬁltration index by one is when it is applied to a linear operator which already has a nontrivial image in F −1 /F 0 (Ad(E)). If this should occur even once, then the net change in the ﬁltration index as a result of applying all the γ i ’s (in the computation of σ) to a is ≤ 0. Thus, if this occurs even once, σ ∈ F 1 (Ad(E)), so tr(σ · [∇, ξ]) = 0. On the other hand, if, during the calculation of σ, we apply α to a linear operator in F 1 (Ad(E)), we get zero. Let us call the case where neither of these two phenomena ever occurs the nondegenerate case. Thus, only the nondegenerate terms τ will make a nonzero contribution. Let us suppose that τ is nondegenerate. Then in order to compute the image of σ in F 0 /F 1 (Ad(E)), it suﬃces to merely keep track of the leading term (relative to the Hodge ﬁltration on Ad(E)) as we apply the various γ i ’s. Now let us note that it follows from the fact that the Kodaira-Spencer morphism for the Hodge ﬁltration on E is the identity that if l ≥ 1, then applying β to a linear operator L in F l (Ad(E)) yields a linear operator in F l−1 (Ad(E)) whose image in F l−1 /F l (Ad(E)) is the image of L in F l /F l+1 (Ad(E)) times ∂. On the other hand, if l ≤ 1, then applying α to a linear operator L in F l Ad(E) yields a linear operator in F l+1 (Ad(E)) whose image in F l+1 /F l+2 (Ad(E)) is the image 65 of L in F l /F l+1 (Ad(E)) times θ SD · ∂. Thus, if τ is nondegenerate, then its contribution −tr(σ · [∇, ξ]) is given by (θ SD · ∂) p · (∂) p . It remains to compute the number of nondegenerate terms τ . Let us call this number N p , and regard it as an element of F p . We are interested in whether or not N p ∈ F p is zero. Although one can presumably compute N p explicitly using some sort of combinatorial argument, we prefer to take the following approach. Note that N p does not depend on g or r or on the particular curve f log : X log → S log , but only on p. Thus, it suﬃces to show that (for each odd prime p) N p = 0 for one particular curve (with g and r arbitrary). We shall do this in §3 when we do various computations with elliptic curves. Thus, we log ⊗2 ) (−D)) F ), when applied to (∂ F ) 2 yields obtain that V E (which is a section of (f ∗ (ω X/S the function c · (θ SD · ∂) p · (∂) p , where c ∈ F × p . Put another way, the degree p component of V g,r is a morphism: [p] log ⊗2 log ⊗2 V g,r : {(f ∗ (ω X/S ) (−D)) ∨ } F → S p (f ∗ (ω X/S ) (−D)) ∨ which is equal to c times the Frobenius morphism, i.e., the p th -power map. We shall see later in our computations with elliptic curves that c = −1. Thus, we thus obtain the following important result: Theorem 2.3. Relative to the aﬃne structures of S g,r and Q g,r , the pull-backs of the aﬃne variables on Q g,r via the morphism V g,r are polynomials in the aﬃne variables of [p] S g,r of degree exactly p, with the leading term V g,r (i.e., the degree p component) given by −1 times the p th -power map. In particular, V g,r is a ﬁnite, ﬂat morphism of degree p 3g−3+r . Proof. It remains to verify the last assertion (that V g,r is ﬁnite and ﬂat of the right degree). Let U = Spec(A) → M g,r be étale. Then over U , we may choose aﬃne coordinates X i and Y j of S g,r and Q g,r so that V g,r looks like (the map induced on Spec’s by): def def B = A[Y 1 , . . . , Y 3g−3+r ] → C = A[X 1 , . . . , X 3g−3+r ] where Y i → f i (X 1 , . . . , X 3g−3+r ), and f i is of the form “−X i p plus terms of lower degree.” Then it is easy to see that, as a B-module, C is generated by monomials of the form 3g−3+r X i e i i=1 where 0 ≤ e i ≤ p − 1. In particular, C is a ﬁnite B-module, so V g,r is ﬁnite. Since V g,r is a ﬁnite morphism between regular algebraic stacks of the same dimension, it follows from commutative algebra that V g,r is ﬂat. To compute the degree of V g,r , let P = Proj A (C[T ]) 66 (with the grading such that T and the X i ’s have degree one). Then Spec(C) ⊆ P is an aﬃne open subset. Let S i ⊆ P be the hypersurface which is the closure of the zero locus of f i . Then the scheme-theoretic intersection V of all the S i ’s has degree p 3g−3+r over Spec(A). Also, the intersection of V with P − Spec(C) = V + (T ) = Proj(C) is just p ) ⊆ Proj(C), which is the empty set. Thus, V ⊆ Spec(C), and so V + (X 1 p , . . . , X 3g−3+r deg(V g,r ) = deg(V → Spec(A)) = p 3g−3+r . Before continuing, we introduce some more terminology that will be of use in the future: Deﬁnition 2.4. Let (E, ∇ E ) be an indigenous bundle on X log . Then, we will say that (E, ∇ E ) is nilpotent if V E is zero. We will say that (E, ∇ E ) is admissible if P E ∨ : Ad(E) → T ∨ is surjective. Note that the terminology of a “nilpotent indigenous bundle” that we have introduced here is (by Proposition 2.1) consistent with that of [Katz]. Also, let us observe that the nilpotent bundles form a closed subscheme N g,r ⊆ S g,r while the admissible bundles form an open subscheme adm S g,r ⊆ S g,r adm adm We shall see later in this Chapter that S g,r N g,r is nonempty and that neither S g,r nor N g,r is contained in the other. Note, further, that Theorem 2.3 implies that the natural morphism N g,r → M g,r is ﬁnite and ﬂat of degree p 3g−3+r . Finally, we observe that one thing which is interesting about nilpotent indigenous bundles is that (if S is the spectrum of a perfect ﬁeld, then) by a result of [Falt] such indigenous bundles arise as the crystalline Dieudonné modules of certain ﬁnite, ﬂat group schemes on X log . This point of view will be pursued further in later chapters. This observation is the main reason for studying N g,r and V g,r , which is the goal of the present Chapter. 67 The p-Curvature of an Admissible Indigenous Bundle As a prélude to ﬁnding out more about V g,r , it is worth looking at various basic properties of the p-curvature of an admissible indigenous bundle. Proposition 2.5. There is a bijective correspondence between nilpotent, admissible in- digenous bundles (up to tensor product with a line bundle of order two) and FL-bundles whose projectivizations are indigenous given as follows: If (E, ∇ E ) is a nilpotent, admissi- ble indigenous bundle on X log , let P E : T → Ad(E) be its p-curvature. Then the kernel of P E ∨ : Ad(E) → T ∨ is an FL-bundle. Moreover, if (E, ∇ E ) is a nilpotent, admissible indigenous bundle on X log , there exists a unique rank one subbundle M ⊆ E that is annihilated by the endomorphisms in the image of P E . This subbundle M is stabilized by ∇ E . The induced connection ∇ M has p-curvature zero, and we have a horizontal isomorphism M ⊗2 ∼ = T . Finally, suppose that S is reduced, and (E, ∇ E ) is nilpotent, admissible, and indige- nous. Let U ⊆ X be an open set, and (L, ∇ L ) a line bundle with logarithmic connection (with respect to f log : X log → S log ) on U . Let ι : L → E| U be a horizontal morphism. Then ι factors through M ⊆ E. Proof. First of all, since (E, ∇ E ) is admissible, P E ∨ is surjective, and thus its kernel is a rank two vector bundle F which is stabilized by ∇ E , hence gets a connection ∇ F . Since (E, ∇ E ) is nilpotent, it follows that T ∼ = Im(P E ) ⊆ F, and that this inclusion T → F is locally split. Also, this inclusion T → F is necessarily horizontal, and we also have a horizontal isomorphism F/T ∼ = O X . Thus, in order to show that (F, ∇ F ) is an FL-bundle, it suﬃces to show that the p-curvature of (F, ∇ F ) is nonzero generically on every ﬁber of f : X → S (by Corollary 1.5). Thus, we may assume that S is the spectrum of a ﬁeld. Now on some nonempty open set U ⊆ X, there is line bundle with connection (L, ∇ L ) and a horizontal surjection μ : E → L such that Im(P E | U ) is given by endomorphisms that vanish on Ker(μ) and whose image is in Ker(μ). Then sorting through what we have done, we see that we have a horizontal isomorphism F| U ∼ = E| U ⊗ L −1 . Since L has p-curvature zero, the fact that E| U has nonzero p-curvature implies that the same is true of F| U . This completes the proof that (F, ∇ F ) is an FL-bundle. On the other hand, if we are given an FL-bundle (F, ∇ F ) whose projectivization (P, ∇ P ) is indigenous, let (E, ∇ E ) be an indigenous vector bundle whose projectivization is (P, ∇ P ). Then there exists a line bundle with connection (L, ∇ L ) (relative to f log : X log → S log ) such that we have a horizontal isomorphism F ⊗ O X L ∼ = E. Taking determinants, ∨ T we thus get a horizontal isomorphism L ⊗2 ∼ , so the p-curvature of L must be zero. = Thus, under the natural identiﬁcation of Ad(F) with Ad(E), we see that the p-curvatures of (E, ∇ E ) and (F, ∇ F ) coincide. Thus, by our computation in Proposition 1.4, (E, ∇ E ) is admissible and nilpotent. Also, it is easy to see that these two procedures are inverse to one another, thus proving the bijective correspondence. We take M ⊆ E to be T ⊗ L ⊆ 68 F ⊗L∼ = E. The remaining assertions follow immediately from what we have done so far, plus Proposition 1.6. Proposition 2.6. Let (E, ∇ E ) be an indigenous bundle on X log . Let P E : T → Ad(E) be its p-curvature. Then: (1) We shall call the composite H E : T → τ X log /S log of P E with the pro- jection E → τ X log /S log arising from the Hodge ﬁltration on E the square Hasse invariant of (E, ∇ E ). If (E, ∇ E ) is admissible, then H E is nonzero. (2) Suppose that (E, ∇ E ) is admissible. Then the zero locus V (H E ) ⊆ X is a divisor D E which is ﬁnite, ﬂat, and of degree (p − 1)(2g − 2 + r) over S. We shall call it the double supersingular divisor of (E, ∇ E ). (3) Suppose that the indigenous bundle (E, ∇ E ) is admissible and nilpotent. Then there exists a line bundle H on X whose square H ⊗2 is isomorphic to T ∨ ⊗ τ X log /S log , together with a section χ of H over X whose square is H E . We shall call χ the Hasse invariant of (E, ∇ E ). In particular, there exists a divisor E E ⊆ X such that D E = 2 E E . We shall call E E the supersingular divisor of (E, ∇ E ). (4) Suppose that S is reduced. Then any two nilpotent, admissible indige- nous bundles with the same supersingular locus are isomorphic. Proof. For (1), it suﬃces to prove the statement after we restrict to a ﬁber of f : X → S; thus, we may assume that S is the spectrum of a ﬁeld. If H E were zero, then that would mean that Im(P E ) lands in F 0 (Ad(E)). Now F 0 (Ad(E)) surjects onto O X . If the image of P E in O X is nontrivial, we get a contradication as follows: On the one hand, Im(P E ) is stabilized by ∇ E . On the other hand, the fact that the Kodaira-Spencer morphism of the Hodge ﬁltration is an isomorphism means that ∇ E applied to Im(P E ) will not be in log F 0 (Ad(E)). If the image of P E in O X is trivial, then it must lie in F 1 (Ad(E)) ∼ = ω X/S . Then, by using the fact that the Kodaira-Spencer morphism is an isomorphism, we again get a contradiction. Assertion (2) follows immediately from (1). Now suppose that (E, ∇ E ) is nilpotent, admissible, and indigenous. Let M ⊆ E be the rank one subbundle of Proposition 2.5. Let E → N be the surjection arising from the Hodge ﬁltration. Then composing the injection M → E with this surjection, we get a morphism χ : M → N , whose square (under the identiﬁcations M ⊗2 ∼ = T ; N ⊗2 ∼ = τ X log /S log ) is equal to the Hasse invariant. Thus, if we let E E be the zero locus of χ, we have D E = 2 E E . This proves (3). To prove (4), we assume that S is the spectrum of a ﬁeld, and that we have two connections ∇ E and ∇ E on E, both of which make E a nilpotent, admissible indigenous bundle, and such that the respective supersingular divisors E E and E E coincide. Let us also 69 assume that ∇ E = ∇ E + θ, where θ is a square diﬀerential. Let ι : M → E and ι : M → E be the respective inclusions, and χ : M → N and χ : M → N the respective composites discussed in the preceding paragraph. We claim ﬁrst of all that M and M are isomorphic. Indeed, this follows from the fact that N ⊗ M −1 ∼ = N ⊗ (M ) −1 . = O X (E E ) = O X (E E ) ∼ Thus, we shall henceforth identify M and M . Now χ and χ diﬀer by multiplication by a section λ of O S , that is, χ = λ · χ . Let i 1 : M → E be ι, and let i 2 : M → E be ι multiplied by λ. Then it follows that there exists a morphism α : M → F 1 (E) such that i 1 = i 2 + α. Now let s be a horizontal section of M (over some open set U ⊆ X). Since the p-curvature is a horizontal morphism, ∇ E (i 1 (s)) = 0 and ∇ E (i 2 (s)) = 0. Thus, we compute: ∇ E (i 2 (s)) = (∇ E + θ)(i 1 (s) + α(s)) = ∇ E (α(s)) + θ(i 1 (s)) Suppose that α is nonzero. Then in the last line, the ﬁrst term has a nonzero image under the surjection E → N (since the Kodaira-Spencer morphism is an isomorphism), while the second term lies in F 1 (E). Since the sum of these terms is zero, we thus obtain a contradiction. Thus, α must be zero. Then we obtain that θ(i 1 (s)) = 0, so (if θ = 0, then) i 1 maps into F 1 (E) ⊆ E. But then χ = 0, so by (2), we again obtain a contradiction. This completes the proof of (4). Proposition 2.7. Suppose that S is reduced. Let (E, ∇ E ) be admissible indigenous on X log . Let P E : T → Ad(E) be the p-curvature. Let U ⊆ X be open, and let (L, ∇ L ) be a line bundle with logarithmic connection (relative to f log ) on U whose p-curvature is zero. Let ι : L → Ad(E) be a horizontal morphism. Then ι factors through T ∼ = Im(P E ). Proof. By shrinking U , we may assume that L and T | U have horizontal generating sections s and t, respectively. We may assume that s and t generate a subbundle G ⊆ Ad(E)| U of rank two. If we take their commutator in Ad(E), we see that [s, t] must be in G. Indeed, if this were not the case, then the p-curvature of Ad(E) would be zero. But the p-curvature of Ad(E) is given by Ad(P E ) which is nonzero everywhere since (E, ∇ E ) is admissible. This proves the claim. Thus, G is a Lie subalgebra of Ad(E), stabilized by the connection on Ad(E) and whose p-curvature is zero. Being of rank two, it is necessarily solvable, hence contains a nilpotent subalgebra K ⊆ G which is stabilized by the connection on Ad(E) and has p-curvature zero. Now K deﬁnes a horizontal ﬁltration N ⊆ E| U with respect to which it is nilpotent. Since (N ) ⊗2 ∼ = K, it follows that N has p-curvature zero. Let δ : End(E| U ) → Ad(E)| U be the canonical projection given by quotienting out by the scalar endomorphisms. Then clearly the image of N ⊗ E| U ⊆ End(E| U ) (i.e., the endomorphisms that anihilate N ⊆ E| U ) under δ is equal to G. Since G and N have p-curvature zero, it thus follows that E| U has p-curvature zero, which contradicts the fact that (E, ∇ E ) is admissible. Proposition 2.8. Suppose that f log is obtained by gluing together various f i log , as in 70 the last subsection of Chapter I, §2. Suppose that (E, ∇ E ) is nilpotent, admissible, and indigenous on X log . Then it is automatically of restrictable type. Proof. The subbundle M ⊆ E (of Proposition 2.5) is stabilized by E and has p-curvature zero. Thus, if we restrict to an irreducible component X i log , the monodromy at any marked point of X i log must be nilpotent with respect to the ﬁltration deﬁned by M ⊆ E. This completes the proof. As mentioned earlier, the reason that we are interested in nilpotent, admissible indige- nous bundles is that they deﬁne MF ∇ -objects in the sense of [Falt]. Let us suppose that S = Spec(k), where k is a perfect ﬁeld, and that f : X → S is smooth. Let S = Spec(A), where A = W (k)/p 2 W (k), and W (k) is the ring of Witt vectors with coeﬃcients in k. Let We suppose that S and S are us denote by Φ A the canonical Frobenius morphism on S. endowed with the trivial log structures, and call the resulting log schemes S log and S log , log → S log be a smooth r-pointed curve of genus g that lifts f log . respectively. Let f log : X ⊆ X be Let (E, ∇ E ) be an indigenous bundle on X log . Let E = F 1 (E) ⊕ (E/F 1 (E)). Let U log → U log be a lifting of Frobenius. If e is a section of E, an open subset, and let Φ log : U Φ ∗ 1 let ∇ E (e) be the section of Φ (E/F (E)) ⊗ ω U log /S log obtained by applying ∇ E to e (and regarding the result modulo F 1 (E) ⊗ ω U log /S log ) to get a section of (E/F 1 (E)) ⊗ ω U log /S log , then pulling back by Φ on E and by p 1 dΦ on ω U log /S log . Then we can deﬁne a logarithmic connection ∇ Φ on Φ ∗ ( E F ) by letting ∇ Φ (Φ −1 (0, e F )) = 0 (if e is a section of E/F 1 (E)) 1 and ∇ Φ (Φ −1 (e F , 0)) = (0, ∇ Φ E (e)) (if e is a section of F (E)). Then, it is easy to see (as in [Falt], §2) that for diﬀerent Φ, the pairs (Φ ∗ ( E F )| U , ∇ Φ ) glue together to form a bundle with connection F ∗ (E, ∇ E ) on X log . Note that F ∗ (E, ∇ E ) depends on the choice of lifting log → S log . X Deﬁnition 2.9. We shall say that (E, ∇ E ) forms an MF ∇ -object on X log if (E, ∇ E ) ⊗ (L, ∇ L ) ∼ = F ∗ (E, ∇ E ) log → S log and some line bundle with connection (L, ∇ L ) whose for some choice of lifting X square is trivial. Note that this deﬁnition is consistent (though slightly weaker, since we allow the ambiguity of tensoring with (L, ∇ L )) with the notion of being an object of the category MF ∇ of [Falt], §2. It is shown in [Falt] (Theorem 6.2) that the de Rham cohomology of a semistable family of varieties over X log equipped with the Hodge ﬁltration and the Gauss-Manin connection forms an object of MF ∇ (as long as p− 2 is greater than or equal to the relative dimension of the family of varieties). The following result provides the link between what we are doing here and [Falt], §2: 71 Proposition 2.10. Let (E, ∇ E ) be an indigenous bundle on X log . Then (E, ∇ E ) forms an MF ∇ -object on X log if and only if (E, ∇ E ) is admissible and nilpotent. Proof. First, let us assume that (E, ∇ E ) forms an MF ∇ -object on X log for the lifting log → S log . One computes easily from the deﬁnition of the connection ∇ Φ that the p- X Also, (just curvature is nilpotent (with respect to the ﬁltration 0 ⊕ (Φ ∗ E/F 1 (E)) ⊆ Φ ∗ E). 1 as in the proof of Proposition 1.4), the derivative p Φ is of the form (1 + t) p−1 + f (t). Thus, since the Kodaira-Spencer morphism is the identity, it follows that the p-curvature def (applied to ∂ = d/dt) is obtained by multiplying a section of (Φ ∗ F 1 (E)) ⊕ 0 by Φ −1 (∂ F ) times the (p − 1) th derivative of (1 + t) p−1 + f (t) (which is just −1) and regarding the result of this multiplication as a section of 0 ⊕ (Φ ∗ E/F 1 (E)). In particular, we see that the p-curvature is nonzero. Thus, (E, ∇ E ) is admissible and nilpotent. On the other hand, suppose that (E, ∇ E ) is admissible and nilpotent. By the bijective correspondence of Proposition 2.5, (E, ∇ E ) corresponds to an FL-bundle (G, ∇ G ). By log → S log . It remains to see Proposition 1.2, this FL-bundle corresponds to a lifting X ∗ that F (E, ∇ E ) taken with respect to this lifting is isomorphic to (E, ∇ E ) (up to tensoring with an (L, ∇ L ) whose square is trivial). Let N = E/F 1 (E). Let M = Φ ∗ X/S (N F ). Let ∇ M be the connection on M for which the sections of N F are horizontal. Thus, the fact that the (M, ∇ M ) ⊗2 ∼ = (T , ∇ T ). Now, sorting through the deﬁnitions and using Kodaira-Spencer morphism is the identity reveals that if X = U V (where U and V are aﬃne opens), then F ∗ (E, ∇ E ) ⊗ (M, ∇ M ) is just the extension of O X by T obtained from the 1-cocycle which is the diﬀerence between Frobenius liftings on U and V . It thus follows from the deﬁnition of the canonical morphism F : D → A of Proposition 1.2 that F ∗ (E, ∇ E ) ⊗ (M, ∇ M ) is exactly the bundle (G, ∇ G ). Thus, it follows from the deﬁnition of the bijective correspondence in Proposition 2.5 that F ∗ (E, ∇ E ) ∼ = (E, ∇ E ) ⊗ (L, ∇ L ) for some (L, ∇ L ) whose square is trivial. The Inﬁnitesimal Verschiebung Let ω X/S be the relative dualizing sheaf of f : X → S. Thus, if D ⊆ X is the divisor of log marked points, we have ω X/S (D) ∼ = ω X/S . Let Φ X/S : X → X F be the relative Frobenius over S. Recall from duality theory (see, e.g., [Harts] for a treatment of duality theory) that since f and Φ X/S are local complete intersection morphisms, we have a trace morphism: tr Φ X/S : (Φ X/S ) ∗ ω X/S → (ω X/S ) F where we regard ω X/S as RΦ ! X/S (ω X/S ) F . On the other hand, let Y log be the log scheme whose underlying scheme is X and whose log structure is the same as that of X log away from the divisor D of marked points, and equal to the pull-back of the log structure of S log on the open subscheme where f is smooth. Then we also have a morphism arising from the log version of the Cartier isomorphism (applied to Y log → S log ): 72 C : (Φ X/S ) ∗ ω X/S → (ω X/S ) F The following result is “well-known,” but I do not know an adequate reference for it: Lemma 2.11. The morphisms tr Φ X/S and C are equal. Proof. By a density argument, we reduce to the case where f is smooth. By naturality, we reduce to the assertion that these two morphisms are the same when X = Spec(F p [T ]); S = Spec(F p ) (where T is an indeterminate). Since tr Φ X/S is the reduction modulo p of a construction that holds in arbitrary or mixed characteristic, we can calculate tr Φ X/S by considering the trace map obtained from duality for the ﬁnite morphism Φ : Spec(Z p [T ]) → Spec(Z p [T ]) given by T → T p . Since tr Φ (Φ ∗ (dT )) = p dT , and Φ ∗ (dT ) = p T p−1 dT , it follows that tr Φ (T p−1 dT ) = dT . By reducing this formula modulo p and comparing with the construction in [Katz] of the Cartier isomorphism, we obtain the desired result. Let (E, ∇ E ) be an indigenous bundle on X log . Let H E : T → τ X log /S log be the square Hasse invariant of (E, ∇ E ). Then by pulling back via Φ X log /S log and taking R 1 f ∗ of H E , we obtain a canonical morphism: Φ τ E : (R 1 f ∗ τ X log /S log ) F → R 1 f ∗ τ X log /S log which we shall call the Frobenius on R 1 f ∗ τ X log /S log induced by (E, ∇ E ). On the other log → T ∨ . Note that hand, let us consider the dual morphism to H E , that is, H E ∨ : ω X/S log F T ∨ = Φ ∗ X/S (ω X/S ) . Thus, if we tensor H E ∨ with ω X/S , we get a morphism log ⊗2 log F ) (−D) → ω X/S ⊗ O X Φ ∗ X/S (ω X/S ) (ω X/S log F If we then compose this morphism with the trace morphism tr Φ X/S tensored with (ω X/S ) , log ⊗2 log ⊗2 we obtain a morphism (Φ X/S ) ∗ (ω X/S ) (−D) → ((ω X/S ) (−D)) F . Then applying f ∗ to this morphism, we obtain: log ⊗2 log ⊗2 (−D) → (f ∗ (ω X/S ) (−D)) F Φ ω E : f ∗ (ω X/S ) log ⊗2 which we shall call the Verschiebung on f ∗ (ω X/S ) (−D) induced by (E, ∇ E ). Observe that by Serre duality applied to the morphism f , we obtain that the vector bundles log ⊗2 ) (−D) and R 1 f ∗ τ X log /S log on S are dual to one another. Then relative to this f ∗ (ω X/S duality, we have τ Proposition 2.12. The morphisms Φ ω E and Φ E are dual to one another. 73 Proof. This follows immediately from duality theory. Namely, the duality between ⊗2 (−D) and R 1 f ∗ τ X log /S log is obtained by cup product, followed by the trace mor- f ∗ ω X/S phism tr f : R 1 f ∗ ω X/S → O S . On the other hand, since trace morphisms behave well under composition, we see that tr f = (tr f ) F ◦ tr Φ X/S . This fact, combined with the fact ∨ that Φ τ E (respectively, Φ ω E ) is obtained from H E (respectively, H E ), implies the result. We are now ready to state and prove the second main result of this Section. Consider the Verschiebung on indigenous bundles deﬁned in the ﬁrst subsection of this Section. In the universal case, it was a morphism V g,r : S g,r → Q g,r over M g,r . Thus, it induces a map on tangent bundles over M g,r : Θ V g,r /M g,r : Θ S g,r /M g,r → Θ Q g,r /M g,r If we pull-back to the point of S g,r deﬁned by our particular (E, ∇ E ), we get a morphism log ⊗2 log ⊗2 Θ V (−D) → (f ∗ (ω X/S ) (−D)) F E : f ∗ (ω X/S ) Then we have the following result: Theorem 2.13. Let (E, ∇ E ) be indigenous on X log . Then the inﬁnitesimal Verschiebung ω morphism Θ V E is equal to Φ E . Proof. First, let us recall Jacobson’s formula (see, e.g., [Jac], pp. 186-187): This formula states that if a and b are elements of an associative ring R of characteristic p, then p p p (a + b) = a + b + p−1 s i (a, b) i=1 where the s i (a, b) are given by the formula: p−1 (ad(ta + b)) (a) = p−1 is i (a, b)t i−1 i=1 computed in the ring R[t], where t is an indeterminate. In our case, we wish to apply this formula in the case where a = α, α ∈ R, and is an element of the center of R such that 2 = 0. In this case, substitution yields: 74 (a + b) p = b p + (ad(b)) p−1 a ω To prove that Θ V E = Φ E , it suﬃces to do a local calculation on X to show that the inﬁnitesimal change in the trace of the square of the p-curvature is given by −2 Φ ω E . We log ⊗2 2 work over the base S[ ]/( ). Let ∇ E = ∇ E + θ, where θ is a section of f ∗ (ω X/S ) (−D), be a connection on E that makes it an indigenous bundle. Let U ⊆ X be an open subset that avoids the marked points and at which f is smooth. Let x be a local coordinate on U . def d . Write ∇ (respectively, ∇ ; θ x ) for ∇ E (respectively, ∇ E ; θ) applied in the direction ∂ = dx We wish to apply the above formula in the case where b = ∇ and a = θ x . We thus obtain that the inﬁnitesimal change in the p-curvature is given by (ad(∇)) p−1 θ x . Now the inﬁnitesimal Verschiebung is obtained by multiplying this term by the constant term and then taking the trace (and multiplying by −1). Put another way, minus the inﬁnitesimal Verschiebung is obtained by applying P E ∨ to (ad(∇)) p−1 θ x . Since P E ∨ is horizontal, it commutes with ad(∇), so we ﬁnd that: (∂) p−1 (P E ∨ θ x ) = (∂) p−1 ((P E ∨ θ) · ∂) log F ) obtained by evaluating minus the inﬁnitesimal Verschiebung is the section of Φ ∗ X/S (ω X/S log ⊗2 (which is a section of Φ ∗ X/S {((ω X/S ) )(−D)} F ) at ∂ F . Thus, to complete the proof of the Theorem, we must show that ∨ F (∂) p−1 ((P E ∨ θ) · ∂) = −Φ −1 X/S {(C ⊗ 1)(P E θ) · ∂ } log F ) . Write P E ∨ θ = ν · ζ, Now let ζ be a horizontal, locally generating section of Φ ∗ X/S (ω X/S where ν is a section of ω X/S . Substituting, we see that it suﬃces to show that: F (∂) p−1 (ν · ∂) · ζ = −Φ −1 X/S (C(ν) · ∂ ) · ζ If we then divide out the ζ’s, we see that really (just as in [Katz2], (7.1.2.6)), we are reduced to proving a simple identity concerning diﬀerentiation in characteristic p. Indeed, if we regard the equation as an identity in ν, both sides are Φ −1 X/S O X F -linear in ν and vanish when ν is exact; thus, we are reduced to proving the identity: (∂) p−1 (x p−1 ) = −C(x p−1 dx) · ∂ F which follows from the deﬁnition of C and the fact that (p − 1)! = −1 in characteristic p. 75 Diﬀerential Criterion for Admissibility We maintain the notation of the previous subsection. In the previous subsection, we computed the relative diﬀerential map of the Verschiebung morphism V g,r over M g,r . In fact, however, with a little more preparation, the same calculation allows us to give an explicit representation of the diﬀerential map of the Verschiebung over F p . Moreover, this explicit representation allows us to give a diﬀerential criterion for an indigenous bundle to be admissible, which is also necessary if the bundle is nilpotent. Consider the aﬃne bundle Q g,r → M g,r . Since by deﬁnition, this bundle is the pull back by the Frobenius morphism on M g,r of a bundle over M g,r , it follows that we get log a natural connection ∇ Q on this aﬃne bundle Q g,r → M g,r . Let Q g,r be the log stack log obtained by pulling back to Q g,r the log structure of M g,r . Next, let us consider the canonical exact sequence of tangent bundles on Q g,r : 0 → Θ Q g,r /M g,r → Θ Q log → Θ M log | Q g,r → 0 g,r g,r Thus, our connection ∇ Q induces a splitting Θ Q log ∼ = Θ Q g,r /M g,r ⊕ Θ M log | Q g,r g,r g,r Now let us consider the “full” inﬁnitesimal Verschiebung, i.e., the morphism induced on tangent bundles by V g,r : Θ V g,r : Θ S g,r → Θ Q g,r On the one hand, we know what the projection of Θ V g,r to Θ M log | Q g,r is. Thus, we would g,r like to compute the projection of Θ V g,r to Θ Q g,r /M g,r . We shall soon see that, in fact, we have already computed this projection as well, in the course of proving Theorem 2.13. Suppose (as in the previous subsection) we have a log scheme S log with an r-pointed stable curve of genus g, f log : X log → S log , and an indigenous bundle (E, ∇ E ) on X log . Then the ﬁrst relative parabolic de Rham cohomology module R 1 f DR,∗ (Ad(E)) (as in Chapter I, Theorem 2.8) is naturally isomorphic to the pull-back to S log (via the classifying morphism for {X log , (E, ∇ E )}) of Θ S log . On R 1 f DR,∗ (Ad(E)), we have a Hodge ﬁltration g,r log ⊗2 0 → f ∗ (ω X/S ) (−D) → R 1 f DR,∗ (Ad(E)) → R 1 f ∗ τ X log /S log → 0 where the surjection in the above exact sequence is exactly the pull-back to S log of the projection Θ S log → Θ M log | S g,r . g,r g,r 76 On the other hand, consider the p-curvature of Ad(E): P E : T → Ad(E). Then by applying “R 1 f DR,∗ ” to the dual of P E , we get a morphism R 1 f DR,∗ (Ad(E)) → R 1 f DR,∗ (T ∨ ) Now by Poincaré duality, we have R 1 f DR,∗ (T ∨ ) ∼ = {R 1 f DR,∗ (T )} ∨ Moreover, we computed R 1 f DR,∗ (T ) in Proposition 1.1. In particular, we have a natural inclusion R 1 f ∗ (τ X log /S log ) F → R 1 f DR,∗ (T ). Thus, if we compose the above morphism induced by P E ∨ with the dual surjection to this natural inclusion, we obtain a natural morphism log ⊗2 ) (−D)) F Θ E : R 1 f DR,∗ (Ad(E)) → (f ∗ (ω X/S Then we have the following result: Theorem 2.14. Let (E, ∇ E ) be indigenous on X log . Then the pull-back to S log of the projection of Θ V g,r to Θ M log is given by the surjection in the Hodge ﬁltration; the pull- g,r back to S log of the projection of Θ V g,r to Θ Q g,r /M g,r is given by Θ E . Proof. It remains to prove the statement about the projection to Θ Q g,r /M g,r . To do this we consider an inﬁnitesimal deformation of (E, ∇ E ) over S log [ ]/( 2 ). But the section of log ⊗2 ) (−D)) F that we obtain can be computed locally on X. Moreover, locally on (f ∗ (ω X/S X, this calculation is exactly the same as that of Theorem 2.13. This proves the result. Corollary 2.15. Let (E, ∇ E ) be indigenous on X log . If Θ E is surjective, then (E, ∇ E ) is log log log admissible. In particular, if the morphism V g,r : S g,r → Q g,r is log étale at the image of log the classifying morphism S log → S g,r for (E, ∇ E ), then (E, ∇ E ) is admissible. Proof. Since being admissible is an open condition, it suﬃces to prove the result when S log is the spectrum of an algebraically closed ﬁeld. If V g,r is log étale, then by Theorem 2.14, Θ E must be surjective. Thus, it suﬃces to prove the ﬁrst statement. Suppose that (E, ∇ E ) is not admissible. Thus, the morphism P E ∨ : Ad(E) → T ∨ has a nonempty zero locus. One can compute the p-curvature explicitly at a marked point (where the monodromy is nonzero and nilpotent); it follows that the zero locus does not contain any marked points. Now there are two possibilities: the zero locus either avoids the nodes or it does not. Let us ﬁrst do the case where the zero locus of P E ∨ avoids the nodes. Since P E ∨ is horizontal, its zero locus must be the pull-back of some closed subscheme via Φ X/S . Thus, 77 in particular, there exists some point x ∈ X (which is neither a marked point nor a F node) such that P E ∨ is zero at Φ −1 X/S (x ) (the scheme-theoretic ﬁber). By the deﬁnition of log ⊗2 ) (−D)) F Θ E , it follows that the image of Θ E lands in the subspace V x of H 0 (X, (ω X/S consisting of sections that vanish at x F . Now by Riemann-Roch on curves, V x cannot be log ⊗2 all of H 0 (X, (ω X/S ) (−D)) F , unless g = 0 and r = 3, or g = 1 and r = 1. This completes the proof (under the assumption that P E ∨ avoids the nodes), except for these two special cases. For g = 0, r = 3, we shall show in the subsection of §3 on totally degenerate curves that the unique indigenous bundle on such a curve is necessarily admissible. (One checks easily that there are no vicious circles in the reasoning.) For g = 1, r = 1, we note that P(E, ∇ E ) is necessarily invariant with respect to the automorphism α given by multiplying by −1. Thus, if we pull-back by the morphism X → X given by multiplying by 2, it is still invariant under α. Hence it descends to the four-pointed curve of genus zero Y log of which X log is a log étale double covering. Let us call this descended bundle P(F, ∇ F ). Thus, P(F, ∇ F ) is indigenous on Y log . It is easy to see that Θ E and Θ F are the same morphism; thus, the hypothesis holds for P(F, ∇ F ), as well. Thus, we reduce to the case g = 0, r = 4, which has already been checked. Now let us consider the case where P E ∨ vanishes at a node ν ∈ X. Let π log : Z log → X log be the partial normalization of X log at ν (where the log structure on Z log is such def that the points mapping to ν are marked points). Let (F, ∇ F ) = π ∗ (E, ∇ E ). Now let us consider the commutative diagram: H 1 (X, τ X log /S log ) F ⏐ ⏐ −→ 1 H DR (X log , Ad(E)) ⏐ ⏐ −→ H 1 (X, τ X log /S log ) ⏐ ⏐ H 1 (Z, τ Z log /S log ) F −→ 1 H DR (Z log , Ad(F)) −→ H 1 (Z, τ Z log /S log ) where the vertical arrows are pull-backs via π; the horizontal arrows on the left are induced by P E ; and the horizontal arrows on the right are induced by the Hodge ﬁltration. Finally, the “prime” on the de Rham cohomology on the bottom row indicates that the we are taking non-parabolic de Rham cohomology on Z log (since (F, ∇ F ) may not even have a natural parabolic structure, if (E, ∇ E ) is not of restrictable type). Let η be a generator of the kernel of H 1 (X, τ X log /S log ) → H 1 (Z, τ Z log /S log ). Let us consider what happens to η F as we move it around the above commutative diagram. Since P E is zero at ν, if we move it 1 (X log , Ad(E)) lies in along the top to H 1 (X, τ X log /S log ), we get zero. Thus, its image in H DR log ⊗2 1 1 ) (−D)). But the pull-back map H DR (X log , Ad(E)) → H DR (Z log , Ad(F)) H 0 (X, (ω X/S log ⊗2 log ⊗2 maps H 0 (X, (ω X/S ) (−D)) injectively into H 0 (Z, (ω Z/S ) ). Going around the other way, however, (i.e., going down and then to the right), we see that the image of η F in 1 1 H DR (Z log , Ad(F)) is zero. We thus conclude that the image of η F in H DR (X log , Ad(E)) is zero. But this contradicts the surjectivity of Θ E , since the upper horizontal morphism on the left-hand side of the above diagram is dual to Θ E . This completes the proof. 78 Conversely, let us suppose that (E, ∇ E ) is nilpotent and admissible. Then we claim that Θ E is necessarily surjective. Indeed, for simplicity, it suﬃces to prove this when S is the spectrum of a ﬁeld. Let K be the kernel of P E ∨ : Ad(E) → T ∨ . Note that the connection on Ad(E) restricts to a connection ∇ K on K. Thus, (K, ∇ K ) is an FL-bundle. Also, we have a horizontal exact sequence: 0 → T → K → O X → 0 in which the connecting morphism O S → R 1 f DR,∗ (T ) must be injective (since (K, ∇ K ) is an FL-bundle). Moreover, it is a tautology that the composite of this morphism O S → R 1 f DR,∗ (T ) with the projection R 1 f DR,∗ (T ) → O S of Proposition 1.1 is the identity. Thus, we see that the morphism R 1 f ∗ (τ X log /S log ) F → R 1 f DR,∗ (T ) → R 1 f DR,∗ (K) is injective. Now let C = Ad(E)/T . We have a connection ∇ C on C, induced by the connection on Ad(E). Thus, we get a horizontal exact sequence: 0 → O X → C → T ∨ → 0 log F ) → R 1 f DR,∗ (O X ) must be injective (since in which the connecting morphism (f ∗ ω X/S otherwise, C would admit two horizontal, generically linearly indendent sections, which contradicts the fact that the p-curvature is nonzero). Lastly, we consider the horizontal exact sequence: 0 → K → Ad(E) → T ∨ → 0 log F ) → R 1 f DR,∗ (K) with the in which the composite of the connecting morphism (f ∗ ω X/S projection R 1 f DR,∗ (K) → R 1 f DR,∗ (O X ) is injective, as we observed above. Since the image of R 1 f ∗ (τ X log /S log ) F ⊆ R 1 f DR,∗ (K) under the map R 1 f DR,∗ (K) → R 1 f DR,∗ (O X ) is zero, it thus follows that if we compose the inclusion R 1 f ∗ (τ X log /S log ) F ⊆ R 1 f DR,∗ (K) with the morphism R 1 f DR,∗ (K) → R 1 f DR,∗ (Ad(E)), the resulting morphism R 1 f ∗ (τ X log /S log ) F → R 1 f DR,∗ (Ad(E)) is injective. But this morphism is dual to Θ E . Thus, Θ E is surjective. This completes the proof of the claim. log log Let N g,r be the log stack obtained by pulling back the log structure on M g,r to N g,r . log Since N g,r is the zero locus of the Verschiebung, it follows that N g,r is log smooth over F p at a point if and only if Θ E is surjective. In other words: Corollary 2.16. Suppose that (E, ∇ E ) is nilpotent indigenous. Then it is admissible if log and only if N g,r is smooth over F p at the image of classifying morphism S log → N g,r for (E, ∇ E ). 79 §3. Hyperbolically Ordinary Curves Often, in the literature, one speaks of a curve as being “ordinary” if its Jacobian is ordinary. In fact, however, since the Jacobian only represents the “abelian part” of the curve, it is in some sense more intrinsic to speak of a curve as ordinary if it is hyperbolically ordinary in the sense deﬁned below. Philosophically, this means that the Verschiebung on indigenous bundles is a local isomorphism in a neighborhood of an indigenous bundle that provides a “nice” uniformization for the curve. Thus, relative to the analogy (explained in the Introduction) between the Verschiebung on indigenous bundles and the Beltrami equa- tion, to be hyperbolically ordinary means that the Verschiebung acts (at least locally) as one might expect from this analogy, given the classical results on existence and uniqueness of solutions to the Beltrami equation. Basic Deﬁnitions Let S log be a ﬁne noetherian log scheme over F p . Let f log : X log → S log be an r-pointed stable curve of genus g (so 2g − 2 + r ≥ 1). Let D ⊆ X be the divisor of marked points. Let (E, ∇ E ) be an indigenous bundle on X log . Let P E : T → Ad(E) be its p-curvature. Recall the Frobenius on R 1 f ∗ τ X log /S log induced by (E, ∇ E ): Φ τ E : (R 1 f ∗ τ X log /S log ) F → R 1 f ∗ τ X log /S log that we deﬁned in §2. Deﬁnition 3.1. We shall call (E, ∇ E ) ordinary if Φ τ E is an isomorphism. Note that the condition of being ordinary is an open condition on S g,r . We shall denote ord this open set by S g,r . Proposition 3.2. If (E, ∇ E ) is ordinary, then it is admissible. Proof. This follows a fortiori from Corollary 2.15. The following deﬁnition is key to the entire paper: Deﬁnition 3.3. We shall say that f log : X log → S log is a hyperbolically ordinary curve if there exists an étale surjection T → S and a nilpotent, ordinary indigenous bundle (E, ∇ E ) on X log × S T . 80 When the context is clear, we shall simply say that f log is an “ordinary curve.” The reason for the descriptive “hyperbolically” is that in the literature, the term “ordinary curve” is frequently used to mean that its Jacobian is ordinary. In this paper, when f log has an ordinary Jacobian, we shall say that f log is parabolically ordinary. Proposition 3.4. If the ﬁber of f log : X log → S log over s ∈ S is hyperbolically ordinary, then there exists an open set U ⊆ S with s ∈ U such that f log | U is hyperbolically ordinary. In particular, f log : X log → S log is hyperbolically ordinary if and only if all its ﬁbers are hyperbolically ordinary. Proof. Indeed, it suﬃces to consider the universal example. Recall that N g,r ⊆ S g,r is the locus of nilpotent indigenous bundles. Write π : N g,r → M g,r for the natural projection. Let n ∈ N g,r ; let m ∈ M g,r be the point π(n). Then it follows from Theorem 2.13, plus the deﬁnition of N g,r as the zero locus of V g,r that if n is ordinary, then π must be étale at n. Thus, π is open at n. This completes the proof. We shall denote the open subscheme of M g,r (respectively, N g,r ) consisting of hyper- ord bolically ordinary curves (respectively, nilpotent, ordinary indigenous bundles) by M g,r ord (respectively, N g,r ). Thus, we have a natural étale surjection ord ord π : N g,r → M g,r ord Finally, let us note that over S g,r , we have an étale local system in F p -vector spaces of dimension 3g − 3 + r obtained by taking the sections of Θ M log | S ord that are invariant g,r g,r under the Frobenius action on Θ M log | S ord given by −Φ τ E . (Note the minus sign in front of g,r g,r −Φ τ E ! It will be important later in Chapter III.) Let us denote this local system by Θ et g,r , ord and call it the tangential local system on S g,r . Similarly, by taking its dual Ω et g,r we obtain ord ord a local system on S g,r which we shall call the diﬀerential local system on S g,r . Often, we shall be interested in the restrictions of these local systems to N g,r . The Totally Degenerate Case In this subsection, we show that totally degenerate curves are hyperbolically ordinary. ord By Proposition 3.4, this will show that M g,r is an open dense subscheme of M g,r . Since totally degenerate curves have no moduli, there is no loss of generality in assuming that S = Spec(F p ). We begin by considering the case g = 0, r = 3. Recall the morphisms constructed at the end of Chapter I, §3: log N log : M 1,1 [2] → M 1,1 81 (parametrizing elliptic curves with level structures on the two-torsion points) and log log Λ log : M 1,1 [2] → M 0,4 which takes an elliptic curve to the four-pointed curve of genus zero of which it is a log double covering. Then in this case, we have M 0,4 = X log . Let us construct a nilpotent, log admissible indigenous bundle on M 0,4 . Since X log has only one indigenous P 1 -bundle (up to isomorphism), this will complete the proof of Corollary 2.15. To do this, we note that (as we saw in Example 3 of Chapter I, §2), the ﬁrst de Rham cohomology module of log the universal elliptic curve over M 1,1 [2] deﬁnes an indigenous vector bundle (E, ∇ E ). Let (P → X, ∇ P ) be the associated P 1 -bundle. Now since the map “multiplication by −1” on an elliptic curve induces the map “multiplication by −1” on E, it follows that (E, ∇ E ) will not descend via Λ. However, since “multiplication by −1” induces the identity on P → X, we see that (P → X, ∇ P ) does descend via Λ. This gives us an indigenous bundle on X log . To see that it is nilpotent and admissible, it suﬃces to see that (E, ∇ E ) is nilpotent and admissible. But by (a rather trivial special case of) [Falt], Theorem 6.2, as a de Rham cohomology module, (E, ∇ E ) necessarily forms an MF ∇ -object (see Deﬁnition 2.9). Thus, Proposition 2.10 tells us that (E, ∇ E ) is nilpotent and admissible. In particular, this completes the proof of Corollary 2.15. Now let us assume that f log : X log → S log is formed by gluing together a number of copies of the 3-pointed stable curve of genus zero (as in the last subsection of Chapter I, §2). Then, as we saw in this ﬁnal subsection of Chapter I, §2, we can glue together the nilpotent, admissible indigenous bundles that we constructed in the previous paragraph to obtain a nilpotent, admissible indigenous bundle (P → X, ∇ P ) on X log . On the other hand, by Proposition 2.8, every nilpotent, admissible indigenous bundle on X log is of restrictable type. Since there is (up to isomorphism) only one indigenous P 1 -bundle of restrictable type on X log , it thus follows that: Proposition 3.5. Up to isomorphism, a totally degenerate r-pointed stable curve of genus g admits one and only one nilpotent, admissible indigenous P 1 -bundle. Next, let us consider the cohomology group H 1 (X, τ X log /S log ) of our totally degenerate curve. If X log is obtained by gluing together various copies X i log of the 3-pointed stable curve of genus zero, let Y log be the disjoint union of the X i log , and let ν log : Y log → X log be the natural map. Let 0 → τ X log /S log → ν ∗ τ Y log /S log → C → 0 be the natural exact sequence (where C is deﬁned so as to make the sequence exact). By considering the long exact cohomology sequence associated to this exact sequence of sheaves, we see that we obtain a natural isomorphism H 1 (X, τ X log /S log ) ∼ = H 0 (X, C). On the other hand, C is naturally isomorphic to the direct sum of the (τ X log /S log )| z , where z ranges over all the nodes in X. Moreover, the residue map gives a natural isomorphism τ X log /S log | z ∼ = O S (well-deﬁned up to sign). Let Σ be the set of nodes of X. Then 82 C = (O S ) z z∈Σ where the subscript “z” is just used as a marker, to indicate which copy of O S one is referring to. Thus, we have a natural isomorphism (well-deﬁned up to sign on each factor): H 1 (X, τ X log /S log ) ∼ = (F p ) z z∈Σ In particular, since F p has a natural bilinear form (given by ring multiplication F p × F p → F p ), using this bilinear form on each factor (F p ) z gives a natural bilinear form: B : H 1 (X, τ X log /S log ) × H 1 (X, τ X log /S log ) → F p which is now independent of all arbitrary choices of sign. Proposition 3.6. For every totally degenerate r-pointed curve of genus g, there is a natural nondegenerate bilinear form B on the O S -module R 1 f ∗ τ X log /S log which takes values in O S . Next we would like to show that the unique nilpotent, admissible indigenous bundle (P → X, ∇ P ) on X log is ordinary. To do this, we must compute the induced Frobenius action on H 1 (X, τ X log /S log ). By using an isomorphism as above H 1 (X, τ X log /S log ) ∼ = (F p ) z z∈Σ we see that it suﬃces to compute the induced Frobenius action on the various (F p ) z ’s. Consider P | z . The Hodge section σ : X → P deﬁnes a point σ z ∈ P | z (F p ). On the other hand, there is a unique point ﬁxed by the monodromy action q z ∈ P | z (F p ). If we think of P as P(J /J [3] ) (as in Chapter I, Proposition 2.5), then (J /J [3] ) z = V σ ⊕ V q , where V σ (respectively, V q ) is the subspace deﬁned by σ z (respectively, q z ). Note that by the residue map, we have natural isomorphisms V σ ∼ = F p and V q ∼ = F p . Thus, we obtain a basis [3] {(1, 0); (0, 1)} of (J /J ) z . Let E σ and E q be the nilpotent endomorphisms of (J /J [3] ) z given, respectively, by the matrices 0 1 0 0 1 0 and 0 0 Thus, E q is essentially the p-curvature of (π : P → X, ∇ P ) restricted to z. Sorting through all the deﬁnitions, it thus follows that the induced Frobenius action on (F p ) z is given by multiplication by tr(E σ · E q ) = 1. We thus obtain the following result: 83 Proposition 3.7. On a totally degenerate r-pointed stable curve X log of genus g over F p , the Frobenius action Φ τP on H 1 (X, τ X log /S log ) induced by the unique nilpotent, admissible indigenous bundle on X log is the identity. In particular, this unique nilpotent, admissible indigenous bundle is ordinary, and so is X log . ord ord Corollary 3.8. The open subschemes M g,r ⊆ M g,r and N g,r ⊆ N g,r are nonempty. The Case of Elliptic Curves: The Parabolic Picture One can get a better feel for ordinariness for general r-pointed stable curves of genus g by ﬁrst studying ordinariness for elliptic curves. In the case of elliptic curves, there are, in fact, two possible theories of ordinary bundles and curves: the parabolic theory and the hyperbolic theory. Indeed, let f log : X log → S log be a 1-pointed stable curve of genus 1, with marked point : S → X. Let Y log be the log scheme obtained from X log (as in the subsection “The Inﬁnitesimal Verschiebung” of §2) by removing the marked point. Then the parabolic theory (respectively, hyperbolic theory) is obtained by considering the various properties of the p-curvature of indigenous bundles on Y log (respectively, X log ). So far in this Chapter, of course, we have only been considering the hyperbolic theory. However, since the notion of an indigenous bundle is deﬁned for Y log , one can consider its p-curvature, and deﬁne the notions of a nilpotent indigenous bundle, or an admissible indigenous bundle, just as before. Also, many of the results (though not all) such as Theorem 2.3 (where we replace the “3g − 3 + r” by 1) continue to hold in the parabolic context. The purpose of this subsection is to summarize what happens when one studies elliptic curves from the parabolic point of view, and to show, in particular, that the notion of ordinariness that we have deﬁned in this paper (in terms of the p-curvature of indigenous bundles) reduces to the classical notion of ordinariness of elliptic curves. log log First, we introduce some notation. Let M 1,0 = M 1,1 . (The point here is that we log shall use the notation M 1,0 when we are thinking about elliptic curves from the parabolic point of view.) Let f : G → M 1,0 be the universal elliptic curve, with identity section : M 1,0 → G. Let L = f ∗ ω G/M 1,0 be the Hodge bundle. Let G log be the log stack whose log underlying stack is G and whose log structure is the pull-back of the log structure on M 1,0 . Let S 1,0 → M 1,0 be the torsor over L ⊗2 of Schwarz structures on G log . Then just as before, we can deﬁne a Verschiebung: V 1,0 : S 1,0 → Q 1,0 Just as before we have a closed subscheme N 1,0 ⊆ S 1,0 consisting of nilpotent indigenous adm bundles, and an open subscheme S 1,0 ⊆ S 1,0 consisting of admissible indigenous bundles. Also, just as before, we deﬁne an indigenous bundle to be ordinary if its inﬁnitesimal Ver- schiebung is an isomorphism, and we deﬁne an elliptic curve to be (parabolically) ordinary if it admits a nilpotent, ordinary indigenous bundle. 84 Now recall that in Example 2 of Chapter I, §2, we constructed a canonical indigenous bundle on G log . This indigenous bundle thus deﬁnes a global section τ S : M 1,0 → S 1,0 which trivializes the L ⊗2 -torsor S 1,0 → M 1,0 . Moreover, by [KM], p. 227, one knows that if p ≥ 5, then H 0 (M 1,0 , L ⊗2 ) = 0, so this trivialization is unique. Let us also recall that, in the deﬁnition of the indigenous bundle (E, ∇ E ) in Example 2 of Chapter I, §2, we had a subbundle 0 ⊕ O G ⊆ ω ⊕ O G = E (where ω = ω G/M 1,0 ) which was stabilized by the connection ∇ E . Moreover, the induced connection on O G was the trivial connection. Put another way, E admits a nonzero horizontal section. It thus follows that the p-curvature of ∇ E is nilpotent. Thus, τ S : M 1,0 → S 1,0 lands inside N 1,0 . Let θ be a section of L ⊗2 over some étale V → M 1,0 . Let ∇ θ E be the connection formed by adding to ∇ E the endomorphism given by E = ω ⊕ O G → O G → ω ⊗2 ∼ = (ω ⊕ 0) ⊗ ω ⊆ E ⊗ ω where the ﬁrst morphism is the natural projection; the second morphism is multiplication by θ; and the ﬁnal inclusion is the natural one. Let δ be a section of L over V which is everywhere nonzero. Thus, the sections δ and 1 deﬁne a global trivialization of E = ω ⊕ O G over G V . We shall write sections of E| V in terms of this basis, given by δ and 1. Write θ = φ · δ 2 , where φ is a function on V . Let ∇ θ be the morphism E| V → E| V given by evaluating ∇ θ E on δ −1 . Then we see that ∇ θ is given by the following matrix: 0 φ 1 0 To obtain the p-curvature of ∇ θ E , we must iterate this matrix p times. This yields the matrix: φ 1 2 (p−1) · 0 φ 1 0 Now let us write δ −p = h · δ −1 (so h is the classical Hasse invariant). Thus, to compute the p-curvature of ∇ θ E , we must subtract from the matrix just given the following matrix: h · 0 φ 1 0 Thus, we obtain that the p-curvature of ∇ θ E is given by: (φ 1 2 (p−1) − h) · 85 0 φ 1 0 If we then take the determinant, we obtain our Verschiebung (applied to (δ F ) 2 ): 1 1 −φ · (φ 2 (p−1) − h) 2 = −φ p + 2h · φ 2 (p+1) − h 2 · φ Let us rewrite this in invariant form. The trivialization τ S of S 1,0 → M 1,0 allows us to write S 1,0 as Spec(⊕ i≥0 L ⊗−2i ). On the other hand, Q 1,0 is given by Spec(⊕ i≥0 L ⊗−2p·i ). Thus, V 1,0 is determined by specifying the morphism of quasi-coherent sheaves: Γ V : L ⊗−2p → L ⊗−2i i≥0 Let us denote the component of Γ V that maps into L ⊗−2i by Γ V : L ⊗−2p → L ⊗−2i . Since [i] L is ample, it follows that Γ V = 0 when i > p (as we saw already in the proof of Theorem 2.3). Let χ ∈ Γ(M 1,0 , L p−1 ) be the Hasse invariant (as in [KM], p. 353). Then we see that we have proven the following result: [i] [i] [i] Theorem 3.9. If i = p, then Γ V is multiplication by −1. If i = 12 (p + 1), then Γ V [i] is multiplication by 2 χ. If i = 1, then Γ V is multiplication by −χ 2 . For all other i, [i] Γ V = 0. In particular, this completes the proof of Theorem 2.3. Corollary 3.10. Geometrically, N 1,0 consists of two irreducible components I 1 and I 2 : One, I 1 , is the section τ S . The other, I 2 , is nonreduced, and (I 2 ) red may be described as follows: In [KM], p. 361, one ﬁnds a description of the Igusa curve Ig(p), with its canonical (Z/pZ) × -action. Then (I 2 ) red is the quotient of Ig(p) by the subgroup {±1} ⊆ (Z/pZ) × . In order to see which nilpotent bundles are ordinary, we must compute the derivative of the Verschiebung map. In terms of the local objects we used in the computation above, we obtain that “dV/dφ” is given by: 1 h(φ 2 (p−1) − h) In particular, if h = 0, then the inﬁnitesimal Verschiebung is identically zero, while if h = 0, then the inﬁnitesimal Verschiebung at τ S is nonzero. Moreover, because of the square factor in the expression for the Verschiebung, we see that if h = 0, then the only nilpotent indigenous bundle at which the inﬁnitesimal Verschiebung will be nonzero is the indigenous bundle given by τ S . We thus obtain the following: Theorem 3.11. An elliptic curve is parabolically ordinary if and only if it is ordinary in the classical sense (i.e., its Hasse invariant is nonzero). If it is ordinary, then the indigenous bundle constructed in Example 2 of Chapter I, §2, is the unique nilpotent, ordinary indigenous bundle on the curve. 86 The Case of Elliptic Curves: The Hyperbolic Picture In this subsection, we consider 1-pointed stable curves of genus 1 as hyperbolic objects. In particular, we shall highlight the numerous contrasts with the parabolic viewpoint presented above. We begin by considering the torsor S 1,1 → M 1,1 . Recall that this torsor has a canonical trivialization at inﬁnity, deﬁned by the unique nilpotent, admissible indigenous bundle on the singular 1-pointed stable curve of genus 1 (Proposition 3.5). Let us suppose that our prime p is ≥ 5. Then it follows from Chapter I, Theorem 3.6, that S 1,1 → M 1,1 does not admit a section which passes through the canonical trivialization at inﬁnity. Now let us consider the closed subscheme N 1,1 ⊆ S 1,1 . By Theorem 2.3, the natural morphism N 1,1 → M 1,1 is ﬁnite and ﬂat of degree p. Let us consider the irreducible component I ⊆ N 1,1 which passes through the canonical trivialization at inﬁnity. Then I is generically reduced. Moreover, the degree of I → M 1,1 must be ≥ 2. This behavior already is substantially diﬀerent from the parabolic case, where the irreducible component passing through the unique nilpotent, ordinary indigenous bundle at inﬁnity has degree one over M 1,0 . Thus, in particular, N 1,1 (respectively, V 1,1 ) is not isomorphic to N 1,0 (respectively, V 1,0 ), despite the fact that as stacks, M 1,1 ∼ = M 1,0 ; S 1,1 ∼ = S 1,0 ; Q 1,1 ∼ = Q 1,0 . Since N 1,1 → M 1,1 has degree p, it follows that there exist points of N 1,1 over the inﬁnity point of M 1,1 at which N 1,1 → M 1,1 is not étale. Such points correspond to nilpotent indigenous bundles which are not admissible (by Proposition 3.5). This fulﬁlls our earlier pledge to show the existence of such bundles. Of these various observations, we record the following for later reference: Proposition 3.12. If p ≥ 5, then the irreducible component of N 1,1 passing through the canonical trivialization at inﬁnity is generically reduced and has degree ≥ 2 over M 1,1 . The Generic Uniformization Number We return to the case of an r-pointed stable curve of genus g, where r and g are arbitrary (but satisfy 2g − 2 + r ≥ 1). Suppose we are given the combinatorial data Δ (as at the end of Chapter I, Secion 2: consisting of a graph Γ, plus λ i ’s, etc.) for a totally degenerate curve. We shall call two collections of such data Δ and Δ equivalent if they deﬁne isomorphic totally degenerate curves. Let us denote by D g,r the equivalence classes of such data Δ. Alternatively, one may think of D g,r as the set of isomorphism classes of totally degenerate r-pointed stable curves of genus g. Now let us consider the morphism: N g,r → M g,r We know that it is ﬁnite, ﬂat, and of degree p 3g−3+r . If Δ ∈ D g,r , consider the irreducible component I Δ of N g,r that passes through the unique nilpotent, admissible indigenous bundle (as in Proposition 3.5) on the curve corresponding to Δ. Then I Δ is generically reduced. Let G Δ be the degree of I Δ over M g,r . We shall refer to G Δ as the generic 87 uniformization number for the data Δ. (The reason for attaching the term “uniformization” to this number will become apparent in later Chapters.) Let G g,r = G Δ Δ∈D g,r Now let us suppose that p is suﬃciently large so that the class Σ (Chapter I, Theorem 3.4) in H 1 (M g,r , Ω M log ) is nonzero. Then we have the following rough result: g,r Proposition 3.13. For g ≥ 3 and p suﬃciently large, the number G Δ is between 2 and p 3g−3+r . Proof. The upper bound follows from the fact that N g,r → M g,r has degree p 3g−3+r . On the other hand, since N g,r → M g,r is ﬁnite, and M g,r is normal, it follows that if I Δ had degree 1 over M g,r , it would, in fact, be isomorphic to M g,r , hence deﬁne a section of S g,r over M g,r . By Chapter I, Theorem 3.4, we know that this is impossible, for p suﬃciently large. It is not clear to the author how far these bounds are from being sharp. For instance, it could be the case that N g,r is, in fact, irreducible. To compute the number G Δ exactly would involve understanding the monodromy around curves that are not hyperbolically ordinary. That is to say, it would involve proving a sort of hyperbolic analogue of Igusa’s theorem on the monodromy around supersingular elliptic curves in the parabolic case. It is interesting to know, however, that G Δ = 1 because this constitutes a depar- ture from the behavior of complex indigenous bundles. To see this, we must ﬁrst explain certain aspects of the analogy between the complex case and the characteristic p case treated here. First of all, the condition of being nilpotent (and ordinary) is analogous, in the complex case, to having real monodromy. Indeed, to be nilpotent (and admissible) is (by Proposition 2.10) the same as coming from an MF ∇ -object in the sense of [Falt]. But to be an MF ∇ -object means, essentially, that the bundle with connection admits a Frobenius action, i.e., that the monodromy is Frobenius-invariant. Since the Frobenius at the inﬁnite prime is complex conjugation, it is thus natural to regard nilpotent (and admissible) bundles as the characteristic p analogue of complex indigenous bundles with real monodromy. On the other hand, in the complex case, within the real-analytic space of complex indigenous bundles with real monodromy, there is a canonical, topologically isolated component, corresponding to the indigenous bundle arising from the uniformiza- tion by the upper half-plane. On the other hand, in the characteristic p case treated here, the fact that I Δ has degree ≥ 2 over M g,r means that there is no canonical choice of a nilpotent, ordinary indigenous bundle, even on a generic curve: since the monodromy at the curves which are not hyperbolically ordinary is nontrivial, one such indigenous bundle is always carried around to another. 88 Chapter III: Canonical Modular Frobenius Liftings §0. Introduction The present Chapter is central to the entire paper. In it, we construct a canonical ord Frobenius lifting on (N g,r ) log , and a canonical indigenous bundle on the universal curve ord over (N g,r ) log . This pair of a canonical Frobenius lifting and a canonical indigenous bundle are uniquely characterized by the fact that, relative to this Frobenius lifting, the renormalized Frobenius pull-back (Deﬁnition 1.4) of the canonical indigenous bundle is equal to itself. In some sense, an ordinary (in the sense of Deﬁnition 1.1) Frobenius lifting is like a p-adic analogue of a Kähler metric on a complex manifold in that it gives rise to local canonical coordinates. Since there are a number of general properties of ordinary Frobenius liftings that we will need throughout the rest of the paper, we give a basic exposition of the properties of such Frobenius liftings in the ﬁrst Section of this Chapter. The main result is that such a Frobenius lifting deﬁnes canonical aﬃne and multiplicative ord coordinates. Thus, in particular, our canonical Frobenius lifting on (N g,r ) log deﬁnes such ord ord canonical coordinates on N g,r . Since N g,r is étale over M g,r , if one thinks of a point ord of N g,r as a point of M g,r , together with a choice (from a ﬁnite number of possibilities) of some added structure – which we call a p-adic quasiconformal equivalence class – then we obtain the result that for every choice of a p-adic quasiconformal equivalence class, we obtain a canonical local uniformization of M g,r . The reason for the name “quasiconformal equivalence class,” is that once one chooses this piece of data for a curve, we shall see in this Chapter and in following Chapters that the uniformization theory of the curve is entirely determined. This is reminiscent of Bers’ approach (as in [Bers]) to proving that hyperbolic curves can be uniformized by the upper half plane: Namely, he proves that (in the complex case) all hyperbolic curves with the same genus and number of marked points belong to the same quasiconformal equivalence class. Thus, once we choose this class in the p-adic case, we obtain a “covariant” uniformization by the aﬃne space modeled on the tangent space to M g,r at the curve in question. To obtain uniformizations by the quadratic diﬀerentials (as in the complex case), we need more information than just the quasiconformal equivalence class. Namely, we need a topological marking of the curve. Once we deﬁne this, we obtain uniformizations by the quadratic diﬀerentials. The canonical Frobenius gives rise to another natural notion, for which I know no parallel in the complex case: a canonical lifting of a curve over a perfect ﬁeld to the ring of Witt vectors with coeﬃcients in that ﬁeld. This is reminiscent of the canonical lifting of an elliptic curve in Serre-Tate theory. In fact, this analogy is more than philosophical: Just as in Chapter II, by using indigenous bundles on elliptic curves – regarded parabolically – one can obtain a similar uniformization theory, involving a canonical Frobenius lifting (on the moduli stack of ordinary elliptic curves) and a canonical indigenous bundle. We then compute that these canonical objects for elliptic curves are precisely the canonical objects 89 that one obtains from classical Serre-Tate theory. Thus, one may regard the theory of uniformizations and canonical liftings discussed in this Chapter as the natural hyperbolic analogue of Serre-Tate theory. §1. Generalities on Ordinary Frobenius Liftings Let k be a perfect ﬁeld of odd characteristic p. Let A = W (k), the ring of Witt vectors with coeﬃcients in k. Let S be a formally smooth, geometrically connected p-adic formal scheme over A of constant relative dimension d. Let S log be a log formal scheme whose underlying formal scheme is S and whose log structure is given by a relative divisor with normal crossings D ⊆ S over A. Let Φ A : A → A be the Frobenius morphism on A. Let us denote the result of base changing by Φ A by means of a superscripted “F .” Let Φ log : S log → S log be a Frobenius lifting, i.e., a morphism whose reduction modulo p is the usual Frobenius morphism in characteristic p. In this Section, we shall study the case of a certain kind of Frobenius lifting, called an ordinary Frobenius lifting. It turns out that such Frobenius liftings deﬁne various types of canonical parameters. It is these canonical parameters that will constitute the various “uniformizations” that we obtain in this paper. Basic Deﬁnitions Let us consider the morphism log dΦ log : Φ ∗ Ω log S/A → Ω S/A induced by Φ log on logarithmic diﬀerentials. It is always divisible by p. Deﬁnition 1.1. We shall call Φ log : S log → S log an ordinary Frobenius lifting if p 1 · dΦ log : log Φ ∗ Ω log S/A → Ω S/A is an isomorphism. We shall use the notation log Ω Φ : Φ ∗ Ω log S/A → Ω S/A to denote the isomorphism p 1 · dΦ log . Note that by considering the sections of Ω log S/A which are invariant under Ω Φ , we obtain an étale sheaf Ω et Φ in free Z p -modules of rank d on S. Deﬁnition 1.2. We shall call Ω et Φ the canonical diﬀerential local system on S associated to Φ log . We shall call its dual Θ et Φ the canonical tangential local system on S associated to log Φ . 90 Moreover, by taking the sections of Ω et Φ to be horizontal, we obtain a natural connection log log ∇ Ω on Ω S/A which is associated to Φ . Note that since Ω et Φ is an étale (not just log étale) local system on S, the connection ∇ Ω is a connection on S with respect to the trivial log structure, i.e., it has no logarithmic poles at D. The Uniformizing Galois Representation Now we would like to associate to Φ log a canonical “uniformizing MF ∇ -object” (where we use the category MF ∇ in the sense of [Falt], §2) as follows. Let P = Ω log S/A ⊕ O S We regard P as being ﬁltered by taking the ﬁltration def def def 0 F 2 (P) = 0; F 1 (P) = Ω log S/A ⊕ 0 ⊆ P; F (P) = P Let ∇ P be the logarithmic connection on P obtained as follows: We start with the connec- tion ∇ P on P which is the direct sum of the trivial connection on O S and the connection ∇ Ω on Ω log S/A . Then we add to ∇ P the End(P)-valued logarithmic diﬀerential given by log log ∼ P → Ω log S/A = (0 ⊕ O S ) ⊗ O S Ω S/A ⊆ P ⊗ O S Ω S/A where the ﬁrst morphism is the projection on the ﬁrst direct summand. The resulting logarithmic connection on P will be called ∇ P . Note that the Kodaira-Spencer morphism for F 1 (P) ⊆ P with respect to ∇ P is the identity map. Next, we deﬁne the Frobenius action on P as follows: We take the Frobenius action to be the morphism P Φ : Φ ∗ P → P which is diagonal with respect to the direct sum decomposition P = Ω log S/A ⊕ O S and is equal to Φ −1 on O S and to dΦ log on Ω log S/A . One sees easily that this Frobenius action P Φ is horizontal with respect to ∇ P . Note that this implies that ∇ P is integrable, since its ∨ 2 log log acts curvature would deﬁne a Frobenius-invariant section of (Ω log S/A ) ⊗ O S ∧ Ω S/A , but Φ on this bundle with slope 1, so any Frobenius-invariant section must necessarily vanish. Thus, (P, F · (P), ∇ P , P Φ ) deﬁnes an MF ∇ -object in the sense of [Falt], §2. Deﬁnition 1.3. We shall call this MF ∇ -object the uniformizing MF ∇ -object on S log associated to Φ log . 91 log Now choose a base point s : Spf(A) → S that avoids D. Let Π S log = π 1 (S K , s K ), where K is the quotient ﬁeld of A, and by the fundamental group of “S log ,” we mean the fundamental group of the open formal subscheme which is the complement of divisors that deﬁne the log structure. Then by the theory of [Falt], §2, the uniformizing MF ∇ -object on S log deﬁnes a dual crystalline Galois representation P et of Π S log . Deﬁnition 1.4. We shall call this Galois representation the uniformizing Galois repre- sentation on S log associated to Φ log . Note that O S ∼ = (0 ⊕ O S ) ⊆ P is stabilized by ∇ P and P Φ , and thus deﬁnes an MF ∇ - subobject of the uniformizing object which is equal to the trivial MF ∇ -object. If we take the quotient of the uniformizing MF ∇ -object by this subobject, we obtain the MF ∇ - object corresponding to the étale Galois representation Θ et Φ , Tate twisted once. Thus, we have an exact sequence of Π S log -modules 0 → Θ et Φ (1) → P et → Z p → 0 which thus deﬁnes an extension class: η Φ ∈ H 1 (Π S log , Θ et Φ (1)) We remark relative to the analogy between Frobenius liftings and Kähler metrics, that the class η Φ formally “looks” somewhat like the diﬀerential form that deﬁnes a Kähler metric. Also, we can deﬁne a ring with Galois action which will be useful later. First consider the symmetric algebra on Ω et Φ (−1) over Z p : S · Z p Ω et Φ (−1) Let us consider the Galois action of Π S log on this symmetric algebra which diﬀers from the direct sum of the actions on the S i (Ω et Φ (−1)) by the class η Φ . Thus, in other words, Spec of the symmetric algebra with this Galois action parametrizes sections of the exact sequence 0 → Θ et Φ (1) → P et → Z p → 0 If we then adjoin the divided powers of the augmentation ideal to this Z p [Π S log ]-algebra, we obtain a Z p -algebra A Φ . Let T log → S log be the ﬁnite covering given by Φ log (so 92 T log ∼ = S log ). Since this ﬁnite covering is log étale in characteristic zero, we may form the subgroup Π T log ⊆ Π S log corresponding to this covering. Then one sees easily that we have a natural Π T log -action on A Φ , compatible with the Π S log -action on the symmetric algebra. (We need to restrict to Π T log so that the Galois action respects the divided powers.) Deﬁnition 1.5. We shall call η Φ the canonical Galois extension class associated to Φ log . We shall call A Φ the ring of additive periods of Φ log . The Canonical p-divisible Group Let us look at the uniformizing MF ∇ -object (P, F · (P), ∇ P , P Φ ) again. Let U ⊆ S be the open formal subscheme which is the complement of the divisor U . Then by [Falt], Theorem 7.1, this MF ∇ -object deﬁnes a p-divisible group G Φ over U . Moreover, just as with the corresponding Galois representation discussed in the previous subsection, we have an exact sequence of p-divisible groups over U : 0 → Θ et Φ (1) ⊗ Z p Q p /Z p → G Φ → Q p /Z p → 0 Deﬁnition 1.6. We shall call G Φ the canonical p-divisible group associated to Φ log . Now let ω ∈ Ω et Φ be an element whose reduction modulo p is nonzero. Thus, ω deﬁnes a Z p -linear morphism ω(−) : Θ et Φ → Z p Let S n → S be the étale covering deﬁned by taking the kernel of the morphism Π S log → n n GL(Θ et Φ ⊗ Z/p Z). Let U n → U be its restriction to U . Then over U n , ω(−) ⊗ Z/p Z will be Galois equivariant, so that, by pushing forward the above exact sequence by means of ω(−), we obtain an exact sequence of ﬁnite ﬂat group schemes: 0 → Z/p n Z(1) → G ω,n → Z/p n Z → 0 which, by Kummer theory, deﬁnes an element n × × p u ω,n ∈ Γ(U n , O U )/Γ(U n , O U ) n n and thus a diﬀerential 93 ω n = (du ω,n )/u ω,n ∈ Γ(U n , Ω U n /A ⊗ Z/p n Z) def Now let n → ∞. Let S be the p-adic completion of the inverse limit of the S n . Since the various ω n are compatible, we thus obtain a diﬀerential , Ω ω ∈ Γ( U /A ) U Now we would like to claim that ω is none other than the original diﬀerential ω that we started out with. In some sense, I believe that this fact is well-known, but I do not know of a clear reference for this fact, so I will prove it explicitly here. First, however, we need to make a few more general observations concerning G Φ . The proof will be given in the subsection after the next. Logarithms of Periods Suppose that k is algebraically closed, and let z : Spf(A) → S be a rational point whose reduction modulo p is equal to the base point s. In particular, it follows that z (since it coincides with the base point s maps into U , and factors canonically through U modulo p). Thus, we can restrict the G ω,n to Spec(A) via z so as to obtain an extension 0 → Q p /Z p (1) → G ω,z → Q p /Z p → 0 of p-divisible groups over Spec(A). By Kummer theory, this extension deﬁnes a unit u ω,z ∈ A × whose image in the residue ﬁeld k is 1. On the other hand, we can consider the Dieudonné crystal E ω of G ω,z . Thus, E ω is a free A-module of rank two with a ﬁltration F 1 (E ω ) ⊆ E ω , and a Frobenius action Φ E : E ω F → E ω . This Frobenius action has a unique subspace E F ⊆ E ω (respectively, E V ⊆ E ω ) on which Frobenius acts with slope zero (respectively, one). Also, E V and F 1 (E ω ) deﬁne the same subspace modulo p. Since F 1 (E ω ) and E/F 1 (E ω ) are naturally isomorphic to A, in the future, we shall identify them with A. Thus, by projection E F → E ω → E ω /F 1 (E ω ) = A, we obtain a natural isomorphism of E F with A, and, dually, a natural isomorphism of E V with A. Finally, since E ω = E F ⊕ E V , we may regard F 1 (E ω ) ⊆ E ω as the graph of an A-linear morphism A = E V → A = E F , which, by means of the various canonical trivializations, gives us an element L ω,z ∈ p · A. Theorem 1.7. We have L ω,z = log(u ω,z ). 94 Proof. Let us denote the sequence of Galois modules which are the p-adic Tate modules of the above exact sequence of p-divisible groups by 0 → W 0 → W → W 1 → 0. Recall the exponential map of [BK], p. 359, Deﬁnition 3.10, 1 exp : F 1 (E ω ) ⊗−2 ∼ (Z p (1)) = A → A × = H Gal 1 where the ﬁrst isomorphism is the trivialization referred to above, and H Gal denotes Galois cohomology with respect to Gal(K/K), where K is the quotient ﬁeld of K. By [BK], p. 359, Example 3.10.1, one knows that this exponential map is equal to the ordinary exponential map deﬁned by the exponential series. Let η 2 = log(u ω,z ); η 1 = exp(η 2 ). Now we diagram-chase. Let us denote by P the (inﬁnite dimensional) Galois module B crys f =1 ⊕ B DR + (notation of [BK]). Applying the exact sequence (1.17.1) of [BK] to the exact sequence of Galois modules 0 → W 0 → W → W 1 → 0, we obtain the following commutative diagram: 0 (W 0 ) H Gal ⏐ ⏐ −→ 0 H Gal (W ) ⏐ ⏐ −→ 0 H Gal (W 1 ) ⏐ ⏐ −→ 1 H Gal (W 0 ) ⏐ ⏐ 0 (W 0 ⊗ P) H Gal ⏐ ⏐ −→ 0 H Gal (W ⊗ P) ⏐ ⏐ −→ 0 H Gal (W 1 ⊗ P) ⏐ ⏐ −→ 1 H Gal (W 0 ⊗ P) ⏐ ⏐ 0 H Gal (W 0 ⊗ B DR ) −→ 0 H Gal (W ⊗ B DR ) 0 −→ H Gal (W 1 ⊗ B DR ) −→ 1 H Gal (W 0 ⊗ B DR ) 0 (W 1 ) which maps via the connecting homomorphism to Now we have an element 1 ∈ H Gal 1 0 1 η 1 ∈ H Gal (W ); since the image of η 1 in H Gal (W 0 ⊗P) is zero, we can consider log(η 1 ) = η 2 . 0 0 On the other hand, 1 ∈ H Gal (W 1 ) maps to an element η 3 ∈ H Gal (W 1 ⊗ P) that dies when 1 (W 0 ⊗ P). Thus we see that η 3 comes from hit with the connecting homomorphism to H Gal 0 0 an element η 4 ∈ H Gal (W ⊗ P) which is unique modulo H Gal (W 0 ⊗ P). Mapping η 4 down 0 0 0 one step to H Gal (W ⊗ B DR ), we get η 5 ∈ H Gal (W ⊗ B DR ) that dies in H Gal (W 1 ⊗ B DR ), 0 0 0 0 hence comes from a unique η 6 ∈ H Gal (W ⊗ B DR )/H Gal (W ⊗ P) = T V . Now it follows from the explicit deﬁnitions of the maps in the sequence (1.17.1) of [BK] that η 6 is precisely L ω,z . On the other hand, it follows from general principles of homological algebra that η 6 = η 2 . This completes the proof. Compatibility of Diﬀerentials Now we return to the issue of showing that ω = ω. Let us begin by observing that ω , we can also be deﬁned as follows. By taking the direct limit of the G ω,n ’s restricted to U obtain an extension of p-divisible groups 0 → Q p /Z p (1) → G ω → Q p /Z p → 0 95 . This, moreover, deﬁnes a Dieudonné crystal (E, ∇ E ) with a ﬁltration F 1 (E) ⊆ E, over U and Frobenius action. In fact, (E, ∇ E ) is obtained from (P, ∇ P ) simply by pulling back 0 → O S → P → Ω S/A → 0 via ω· : O U → Ω S/A | U . Now F 1 (E) and E/F 1 (E) may be identiﬁed with O U . Thus, ω is precisely the diﬀerential obtained by considering the Kodaira-Spencer morphism F 1 (E) = O U → Ω U /A ⊗ (E/F 1 (E)) = Ω U /A Now let R z be the complete local ring which is the completion at z of U . Let R z PD be the p-adic completion of the PD-envelope of R z at the augmentation ideal R z → A deﬁned by z. Now taking the inverse limit of the u ω,n ’s deﬁnes a unit u ω ∈ (R z PD ) × whose image in the residue ﬁeld k is 1. Thus, we can consider log(u ω ) ∈ R z PD . On the other hand, let P z = P(z ∗ E) = P(E ω ). Let σ F : S → P z (respectively, σ V : S → P z ) denote the section determined by the subspace E F ⊆ E ω (respectively, E V ⊆ E ω ). The trivializations discussed previously deﬁne an isomorphism of the tangent space to P z at σ F with A. Thus, in summary, we get an isomorphism ψ : P z ∼ = P 1 by sending σ F (respectively, σ V ) to inﬁnity (respectively, zero) and using the trivialization of tangent space to σ F to remove the remaining multiplicative ambiguity. Let P R z PD = P(E)| Spf(R z PD ) . Then ∇ E gives an isomorphism Ξ P : P R z PD ∼ = R z PD ⊗ A P z which, when composed with ψ, gives an isomorphism μ : P R z PD ∼ = P 1 R z PD . Now by Theorem 1.7, it follows that the Hodge section (deﬁned by F 1 (E) → E) σ : Spf(R z PD ) → P R z PD ∼ = P 1 R z PD is (in terms of the standard coordinate t on P 1 , which vanishes at zero and has a pole at inﬁnity) simply log(u ω ). (Indeed, Theorem 1.7 tells us that this is true after restriction to any A-valued point of R z PD ; hence it must be true over R z PD .) It thus follows that the def pull-back of the diﬀerential dt on P 1 via σ is simply ω = du ω /u ω . But, tracing through all the deﬁnitions, the pull-back of dt via the Hodge section is exactly the Kodaira-Spencer morphism of the Hodge ﬁltration. Thus, we conclude that ω = ω over R z PD . On the other (since O → R PD is injective). hand, it is clear that this implies that ω = ω over all of U z U Thus, we have proven the following . Theorem 1.8. We have ω = du ω /u ω = ω over U 96 Note that this holds (by descent) without the assumption that k is algebraically closed. Canonical Liftings of Points in Characteristic p Let α 1 ∈ S(A) be an A-valued point of S. Suppose we apply Φ to α 1 to obtain an A-valued point β 1 ∈ S(A). Then since Φ is a Frobenius lifting, it induces zero on the morphism on cotangent spaces modulo p. Thus, β 1 (mod p 2 ) depends only on α 1 (mod p). Let α 2 = Φ −1 A (β 1 ). Thus, α 2 ≡ α 1 (mod p), and α 2 depends only on α 1 (mod p). If we then continue in this fashion, deﬁning α i+1 = Φ −1 A Φ(α i ) def it is clear that α i ≡ α 1 (mod p) for all i ≥ 1, and that the sequence {α i } of points in S(A) converges p-adically. Let α ∞ ∈ S(A) be the limit of this sequence. Let α 0 ∈ S(k) be the reduction of α 1 modulo p. Note that we have F Φ(α ∞ ) = α ∞ and that, moreover, α ∞ is the unique A-valued point of S which has this property and is equal to α 0 modulo p. Deﬁnition 1.9. We shall call α ∞ the canonical lifting of α 0 . We shall call an A-valued point of S which is a canonical lifting of some k-valued point a canonical A-valued point of S. All the canonical extensions that we have deﬁned become trivial when restricted to α ∞ . More precisely, Proposition 1.10. If α 0 ∈ U (k), then the restriction of u ω to α ∞ is 1. Proof. Indeed, the Hodge ﬁltration of P is invariant under Φ, so its restriction to α ∞ is still Frobenius invariant. By the theory of ﬁltered Dieudonné modules with Frobenius action over A = W (k), it thus follows that the extension of p-divisible groups that one obtains is trivial. Thus, by Kummer theory, u ω | α ∞ = 1. Canonical Multiplicative Parameters Let us assume just in this subsection that k is algebraically closed. Let z ∈ S(k) be a k-valued point of S. Let S z log be the completion of S log at z. Thus, S z log is Spf of a complete local ring R z which is noncanonically isomorphic to 97 A[[t 1 , . . . , t d ]] with the restriction of the divisor D deﬁned by t 1 · t 2 · . . . · t i (where i may be zero). Then log if we restrict Ω et Φ to S z , we obtain the trivial local system. Let ω ∈ Ω et Φ have integral residues at all the irreducible components of D, and nonzero reduction modulo p. Thus, ω deﬁnes a surjection ω(−) : Θ et Φ → Z p If we apply ω(−) to our canonical extension of p-divisible groups 0 → Θ et Φ (1) ⊗ Z p Q p /Z p → G Φ → Q p /Z p → 0 def we obtain an extension of Q p /Z p by Q p /Z p (1) over U z = S z | U . By Kummer theory, we thus obtain a “logarithmic unit” q ω,z ∈ R z [ 1 1 , . . . , ] × t 1 t i which is well-deﬁned up to multiplication by a Teichmüller representative of an element of k. If ω has residue e j at the component of D deﬁned by t j , then the valuation of q ω,z at (t j ) is equal to e j . Indeed, this follows from the formula dq ω,z /q ω,z = ω (of Theorem 1.8). Next, let us consider Φ −1 (q ω,z ). Since Φ −1 multiplies dq ω,z /q ω,z = ω, as well as the p , for some canonical extension of p-divisible groups by p, it follows that Φ −1 (q ω,z ) = λ · q ω,z λ ∈ [k × ] (where the brackets mean “the Teichmüller representative of”). On the other hand, because Φ is a Frobenius lifting, reducing modulo p shows that λ = 1. Thus, we have that p Φ −1 (q ω,z ) = q ω,z Deﬁnition 1.11. We shall call q ω,z for such an ω a canonical multiplicative parameter associated to Φ log . Canonical Aﬃne Coordinates In this subsection, k need not be algebraically closed. Let α ∈ U (A) be canonical. Let A α be the p-adic completion of the PD-envelope of S at the subscheme Im(α). Let α → A be the augmentation that deﬁnes the point α. Let I = Ker( α ). The α : A A-algebra structure, together with the augmentation α deﬁne a splitting 98 A α = A ⊕ I which we shall call the augmentation splitting of A α . Note that we have an A-linear Frobenius action Φ A : (A α ) F → A α induced by the Frobenius lifting Φ. Moreover, Φ A preserves the augmentation splitting, as well as the ring structure of A α . Let us consider the slopes of this Frobenius action Φ A . Clearly, Φ A acts on A⊕0 ⊆ A α as Φ A . Next, we note that since Φ A is a Frobenius lifting, it maps I into p · I. Thus, we have Φ A (I [j] ) ⊆ p j · I [j] where the superscript in brackets denotes the divided power. By the deﬁnition of an def ordinary Frobenius lifting, Ω α = I/I [2] has constant slope one. Thus, if we divide Φ A restricted to Ω α by p, we obtain an isomorphism (Ω α ) F → Ω α Next, let us consider the A-submodule Ω can ⊆ I which is the closure (in the p-adic topology) of the intersection of the images of ( 1 p · Φ A | I ) N (for all N ≥ 1). Since I/I [2] has slope 1, it is clear that the projection Ω can → Ω α is surjective. Now let us consider the intersection of Ω can with I [2] . Let φ = ( p 1 · Φ A | I ) N (ψ), where ψ ∈ I. If φ is contained in I [2] modulo p N , then since I/I [2] has slope 1, it follows that ψ must also be contained in I [2] modulo p N . But then φ = ( p 1 ·Φ A | I ) N (ψ) must be zero modulo p N . Thus, we conclude that the projection Ω can → Ω α must be an isomorphism. Inverting this isomorphism, we thus get a canonical morphism κ A : Ω α → A α Let S α be the formal scheme which is the p-adic completion of the PD-envelope in S of the image of α ∈ U (A). Let Θ α be the dual A-module to Ω α . Let Θ aﬀ α be the p-adic completion of the PD-envelope at the origin of the aﬃne space modeled on Θ α . Thus, Θ aﬀ α is Spf of the p-adic completion of the PD-envelope of the symmetric algebra (over A) of Ω α at the augmentation ideal. We may then reinterpret the canonical morphism κ A as an isomorphism α U can : Θ aﬀ α → S 99 We thus see that we have proven the following result: Theorem 1.12. For every choice of a canonical α ∈ U (A), we obtain a local uniformiza- tion (canonically associated to Φ) ∼ α U can : Θ aﬀ α = S of S by the aﬃne space modeled on Θ α . Deﬁnition 1.13. We shall call the elements of the image of κ A canonical aﬃne parameters associated to Φ at α. Now let (B, m B ) be a local ring with residue ﬁeld k which is p-adically complete and has a topologically nilpotent PD-structure on m B . Let β ∈ S(B) be such that β(mod m B ) ∈ S(k) is equal to α(mod p) ∈ S(k). Let S β be the p-adic completion of the PD-envelope of S ⊗ A B at the image of β in S(B). Thus, S β ∼ = S α ⊗ A B. Let B β = A α ⊗ A B. By tensoring the canonical morphism κ A constructed above with B, we thus obtain a morphism def κ B : (Ω α ) B = (Ω α ) ⊗ A B → B β Let β : B β → B be the augmentation corresponding to the point β ∈ S(B). Let I β = Ker(β). Let B β = B ⊕ I β be the splitting deﬁned by β and the B-algebra structure on B β . Let us consider the projections of κ B on these two factors: κ 0 B : (Ω α ) B → B; κ 1 B : (Ω α ) B → I β [2] Let Ω β = I β /I β . If we compose κ 1 B with the projection to Ω β , we thus obtain a morphism Ψ αβ : (Ω α ) B → Ω β which is an isomorphism, since it is an isomorphism modulo m B , where β coincides with α(mod p). Let Θ β the dual B-module to Ω β . Now we may regard κ 0 B as an element −1 of (Θ α ) B ; if we apply (Ψ ∨ , then we get an element of κ β ∈ Θ β , hence ∈ m B · Θ β αβ ) (since β ≡ ( α ) B modulo m B ). On the other hand, if we compose κ 1 B with Ψ −1 αβ , we get a β morphism Ω β → B which gives us an isomorphism 100 β U β : Θ aﬀ β → S In summary, we have proven the following result: Theorem 1.14. For every β ∈ S(B) as above, we obtain a canonical class κ β ∈ m B · Θ β , as well as a local uniformization ∼ β U β : Θ aﬀ β = S of S by the aﬃne space modeled on Θ β . Moreover, this uniformization is related to the canonical uniformization by tensoring over A with B, applying the isomorphism −1 : (Θ α ) B → Θ β (Ψ ∨ αβ ) and then translating by κ β . Finally, for all β ∈ S(B) whose reductions modulo m B are equal to α(mod p), the correspondence β → κ β is a bijection of such β onto m B · Θ β . Proof. All the statements except the last follow from the way we constructed the ob- jects involved. The last statement about the bijection follows from simply evaluating the canonical uniformization of Theorem 1.12 on B-valued points. Finally, we remark (relative to the analogy between Frobenius liftings and Kähler metrics) that these canonical aﬃne parameters are like canonical coordinates for a real analytic Kähler metric. The Relationship Between Aﬃne and Multiplicative Parameters Let us continue with the notation of the preceding subsection, but let us assume in addition that k is algebraically closed, so that the canonical multiplicative parameters are deﬁned. Let ω ∈ Ω et Φ have integral residues at all the irreducible components of D, and nonzero reduction modulo p. Since Ω et Φ ⊆ Ω α , we may also regard ω as an element of Ω α . Then we would like to establish the relationship between the canonical multiplicative parameter q ω,α and the canonical aﬃne parameter κ A (ω). First note that by Proposition 1.10, q ω,α evaluated at α is a Teichmüller representative. Thus, log(q ω,α ) ∈ A α is zero at α, as is κ A (ω) ∈ A α . Moreover, d log(q ω,α ) = ω, by Theorem 1.8. On the other hand, the fact that d κ A (ω) = ω is a tautology. Thus, log(q ω,α ) and κ A (ω) have the same derivative and both vanish at α. We thus obtain the following result: 101 Theorem 1.15. We have κ A (ω) = log(q ω,α ) in A α . §2. Construction of the Canonical Frobenius Lifting log In this Chapter, we shall denote by M g,r the p-adic formal stack of r-pointed stable curves of genus g over Z p . We shall denote the reductions of objects over Z p to F p or Z/p 2 Z ord by means of a subscripted F p or Z/p 2 Z. Let (N g,r ) F p → (M g,r ) F p be the étale morphism ord in characteristic p of Chapter II, §3. Thus, (N g,r ) F p parametrizes pairs consisting of an r-pointed stable curve of genus g in characteristic p, together with an ordinary, nilpotent ord indigenous bundle. Let N g,r → M g,r be the unique étale morphism of p-adic formal ord ord schemes that lifts (N g,r ) F p → (M g,r ) F p . Thus, N g,r is a smooth p-adic formal scheme ord over Z p whose reduction modulo p is (N g,r ) F p . Our goal in this Section is to construct a ord canonical ordinary lifting of Frobenius on N g,r . Modular Frobenius Liftings In this subsection, we reinterpret certain constructions from Chapter II, §1, in terms of liftings of the Frobenius morphism on (M g,r ) F p . Let S → (M g,r ) F p be étale, and let S log log be the log scheme obtained by pulling back the log structure of M g,r . Thus, in particular, S is smooth over F p . Let Φ S log : S log → S log be the absolute Frobenius morphism. Since (M g,r ) F p ⊆ (M g,r ) Z/p 2 Z is deﬁned by a nilpotent ideal, the étale morphism S → (M g,r ) F p lifts naturally to an étale morphism S → (M g,r ) Z/p 2 Z . We let S log be the log scheme obtained by pulling back the log structure of (M g,r ) Z/p 2 Z . We shall call a Frobenius lifting on S log a log morphism S log → S log whose reduction modulo p is equal to Φ S log . Note that by assigning to étale morphisms U → S the set of Frobenius liftings log , we obtain a sheaf L on the étale site of S, with the natural structure of a torsor on U def ∗ over Θ Φ S log = Φ S Θ S log , where Θ S log is the dual vector bundle to the sheaf of logarithmic diﬀerentials on S log . Moreover, this torsor naturally admits a connection ∇ L as follows: Consider the sheaf of bianalytic functions O S bi on S log . The image I Φ S of the Frobenius −1 morphism Φ S bi : O S bi → O S bi is equal to i L (Φ −1 S O S ), as well as to i R (Φ S O S ) (where i L , i R : O S → O S bi are the left and right injections). Thus, the pull-back of the sheaf L by either i L or i R is equal to the sheaf of liftings of I Φ S to a Z/p 2 Z-ﬂat subalgebra of O S bi . This gives a connection ∇ L on the Θ Φ S log -torsor L → S which is compatible with Φ the natural connection on Θ S log (for which sections of Φ −1 S Θ S log are horizontal); also, one checks easily that ∇ L is integrable. Now let us recall the Θ Φ S log -torsor D → S that we deﬁned in Chapter II, §1. Let log f log : X log → S log be the pull-back to S log of the universal curve over (M g,r ) F p . Recall 102 log F then that D is the Θ Φ ) = X log × S log ,Φ S log S log -torsor consisting of liftings of the curve (X S log → S log to an S-ﬂat curve Y log → S log . Note that it follows immediately from the log deﬁnition of the classifying log stack M g,r (plus the fact that S → M g,r is étale) that we have an isomorphism α : D→L log → S log (which of Θ Φ S log -torsors given by considering the classifying map of the lifting Y is, by deﬁnition, a Frobenius lifting on S log ). On the other hand, the theory of Chap- ter II, §1, gives a natural connection ∇ D on D → S as follows. Recall the line bundle def T = (Φ X log /S log ) ∗ (τ X log /S log ) F on X log . By declaring the sections of the τ X log /S log inside the deﬁnition of T to be horizontal, we see that T gets a natural connection ∇ T over X log (i.e., not just in the relative sense for f log : X log → S log ). Thus, the de Rham cohomology module R 1 f DR,∗ (T ) has a Gauss-Manin connection ∇ GM on S log . By Chapter II, Propo- sition 1.1, we have a natural surjection R 1 f DR,∗ (T ) → O S , which one veriﬁes easily to be horizontal. Since, by Chapter II, Proposition 1.2, D is just the sheaf of sections of this surjection R 1 f DR,∗ (T ) → O S , it thus follows that, as such, D gets a natural connection (induced by ∇ GM ), which we shall call ∇ D . Proposition 2.1. The isomorphism α is horizontal with respect to ∇ D and ∇ L . Proof. Let us denote by X L (respectively, X R ) the pull-back of f log : X log → S log via i L : O S → O S bi (respectively, i R : O S → O S bi ) to S bi . Thus, we obtain a diagram over S bi : X L ← X bi → X R Let us denote the left-pointing (respectively, right-pointing) arrow by π L (respectively, π R ). Similarly, we have an analogous diagram with tildes, for the various objects over Z/p 2 Z. Now consider the image of Frobenius I Φ S ⊆ O S bi . We also have the image of the absolute Frobenius on X, which we denote by I Φ X ⊆ O X bi . Note that I Φ X actually sits inside both O X L and O X R . Suppose next that we are given a Z/p 2 Z-ﬂat subalgebra I Φ S ⊆ O S bi that lifts I Φ S . This corresponds to a section η of L L = L R . (Here the superscript “L”’s and “R”’s denote left and right pull-backs to S bi , respectively.) The obstruction to lifting I Φ X to a Z/p 2 Z- 1 L L S ﬂat subalgebra of O X L compatible with I Φ deﬁnes a class in R (f ) ∗ (T ), which is, by def def deﬁnition, equal to ξ[L] = (α L ) −1 (η). Similarly, we obtain a class ξ[R] = (α R ) −1 (η). Note that (π L ) −1 (ξ[L]) = (π R ) −1 (ξ[R]), since both classes are the obstruction to lifting I Φ X to a Z/p 2 Z-ﬂat subalgebra of O X bi compatible with I Φ S . Let us call this common class ξ[bi]. 103 Now suppose that ( I ) S Φ ⊆ O S is a Z/p 2 Z-ﬂat lifting of I Φ S . Suppose that this lift- ing corresponds to a section ζ of L. If we then take η = ζ L (in the previous para- graph), we get ξ[bi] = (π L ) −1 {(α −1 (ζ)) L }, and, similarly, if we take η = ζ R , we get ξ[bi] = (π R ) −1 {(α −1 (ζ)) R }. On the one hand, ∇ D (α −1 (ζ)) is computed by subtracting these two ξ[bi]’s. On the other hand, (by the deﬁnition of the Θ Φ S log -torsor structure on D) the diﬀerence between these two ξ[bi]’s is the diﬀerence between the two classifying morphisms given by the subalgebras {( I ) S Φ } L and {( I ) S Φ } R of O S bi . But this diﬀerence is, by deﬁnition, ∇ L (ζ). This completes the proof. Henceforth, we shall identify (D, ∇ D ) with (L, ∇ L ), and call the resulting torsor with connection (D, ∇ D ) (since the notation L is more natural for line bundles). Indigenous Sections of D We continue with the notation of the previous subsection. Thus, S → M g,r is étale, and we have the Θ Φ S log -torsor D → S, with its connection ∇ D . Let π : D → S be the scheme corresponding to this torsor. Thus, D is a twisted version of Spec of the symmetric algebra of the dual of Θ Φ S log . We endow D with the log structure pulled back from S; this log gives us a log stack D . On D, taking the dual to the second fundamental exact sequence for diﬀerentials gives a sequence of tangent bundles: 0 → Θ Φ S log | D → Θ D log → Θ S log | D → 0 where Θ D log is the logarithmic tangent bundle on D log . We shall denote the surjection Θ D log → Θ S log | D by Θ π . The connection ∇ D then deﬁnes a connection ∇ D on the ﬁber bundle π : D → S, hence a section ∇ Θ : Θ S log | D → Θ D log of Θ π . Now let us suppose that we are given a section σ : S → D of π. Then σ induces a section of σ ∗ Θ π , which we denote by Θ σ : Θ S log → σ ∗ Θ D log . Also, σ deﬁnes an FL-bundle (E, ∇ E ) (see Chapter II, §1) on the curve X log . Deﬁnition 2.2. We shall call the section σ indigenous if the projectivization of the FL-bundle on X log deﬁned by σ is an indigenous bundle on X log . Let us assume that σ is indigenous. Then we obtain, for i = 1, 2, canonical morphisms of vector bundles φ i : σ ∗ Θ D log → Θ S log deﬁned functorially as follows. By means of the étale morphism S → M g,r , we can think of the geometric vector bundle σ ∗ Θ D log on S as parametrizing inﬁnitesimal deformations η = {(X log ) , (E , ∇ E )} of the curve plus FL-bundle pair given by η = {X log , (E, ∇ E )}. 104 Then the obstruction to lifting the Hodge ﬁltration of (E , ∇ E )| X = (E, ∇ E ) (which exists since σ is indigenous) to a ﬁltration of E over X deﬁnes a section of Θ S log , which we take for φ 2 (η ). On the other hand, if we think in terms of crystals, then (E , ∇ E ) also deﬁnes a deformation (E , ∇ E ) of (E, ∇ E ) on X[ ]/( 2 ). The obstruction to lifting the Hodge ﬁltration of (E, ∇ E ) to a ﬁltration of E on X[ ]/( 2 ) deﬁnes a section of Θ S log , which we take for φ 1 (η ). Since, σ ∗ Θ π (η ) is simply the diﬀerence between (X log ) and the trivial deformation of X log , we thus see that σ ∗ Θ π = φ 2 − φ 1 We also have that φ 1 ◦ σ ∗ ∇ Θ = 0 Indeed, sorting through the deﬁnitions, one sees that the image of ∇ Θ consists of the η = {(X log ) , (E , ∇ E )} obtained by letting (E , ∇ E ) be the FL-bundle given by regarding (E, ∇ E ) as a crystal and taking the bundle with connection that this crystal induces on the deformation (X log ) . Thus, (E , ∇ E ) is simply the trivial deformation of (E, ∇ E ), hence is indigenous on X log by assumption. Putting the above two formulas together, we thus obtain that φ 2 ◦ σ ∗ ∇ Θ = id Θ S log Also, let us note that φ 2 ◦ Θ σ = 0 since if it were nonzero, it would measure exactly the extent to which σ fails to stay within the indigenous locus of D, but, by assumption, σ does stay within the indigenous locus. Next, let us recall the morphism Φ τ E : Θ Φ S log → Θ S log , i.e., the dual to the “inﬁnitesimal Verschiebung” of Chapter II, §2. Recall that this morphism was constructed by applying R 1 f ∗ to the morphism T → τ X log /S log given by composing the p-curvature P : T → Ad(E) of E with the projection Ad(E) → τ X log /S log arising from the Hodge ﬁltration. It thus follows immediately from the deﬁnitions (by thinking about how one deﬁnes the obstruction ∗ that φ 1 measures) that if we restrict φ 1 : σ ∗ Θ D log → Θ S log to Θ Φ S log ⊆ σ Θ D log , we get φ 1 | Θ Φ S log = Φ τ E So far we have been thinking about morphisms that one can obtain from σ by thinking about the indigenous FL-bundle (E, ∇ E ) that it deﬁnes. But by what we did in the previous subsection, σ also deﬁnes a Frobenius lifting Φ σ : S log → S log . Let us consider the 105 morphism Θ Φ σ : Θ S log → Θ Φ S log obtained by looking at the morphism induced by Φ σ on the tangent bundles, and then dividing by p. On the other hand, the morphism Θ σ −σ ∗ ∇ Θ : ∗ Θ S log → σ ∗ Θ D log maps into Θ Φ S log ⊆ σ Θ D log . Thus, by abuse of notation, we shall regard ∗ Θ σ − σ ∇ Θ as a morphism Θ S log → Θ Φ S log . Then we claim that Θ Φ σ = Θ σ − σ ∗ ∇ Θ Indeed, if we think of σ ∗ ∇ Θ as deﬁning a direct sum splitting of σ ∗ Θ D log , then Θ σ − σ ∗ ∇ Θ ∗ is just the component of Θ σ that sits in the vertical subspace Θ Φ S log ⊆ σ Θ D log . Put another ∗ Φ way, Θ σ −σ ∇ Θ : Θ S log → Θ S log is the Kodaira-Spencer morphism for the section σ relative to the connection ∇ D . Thus, it follows from the deﬁnition of the connection called ∇ L in the previous subsection in terms of subalgebras of O S bi that Θ Φ σ = Θ σ − σ ∗ ∇ Θ . We are now ready to prove the main technical result of this subsection: Proposition 2.3. If σ : S → D is an indigenous section, then −Φ τ E is inverse to Θ Φ σ . In particular, the indigenous bundle associated to (E, ∇ E ) is ordinary. Proof. Indeed, using the various observations made above, we simply compute: Φ τ E ◦ Θ Φ σ = φ 1 ◦ (Θ σ − σ ∗ ∇ Θ ) = φ 1 ◦ Θ σ = (φ 1 − φ 2 ) ◦ Θ σ = −(σ ∗ Θ π ) ◦ Θ σ = −id Θ S log Thus, in particular Φ τ E is an isomorphism, and so (E, ∇ E ) deﬁnes an ordinary nilpotent indigenous bundle. Frobenius Invariant Indigenous Bundles ord In this subsection, we change notation slightly. Let S → N g,r be an étale morphism ord of a p-adic formal scheme S into N g,r . Thus, S is formally smooth over Z p . Also, one may ord ord think of S → N g,r as the unique étale lifting of its reduction S F p → (N g,r ) F p modulo p. For convenience, we assume that S F p is aﬃne. Endow S with the log structure pulled log back from M g,r . Thus, we get a p-adic formal log scheme S log . Pulling back the universal curve over M g,r , we get a morphism f log : X log → S log . Let h log : Y log → S log be an log r-pointed curve of genus g whose reduction modulo p is equal to (X log ) F F p → S F p , i.e., the Frobenius transform of f F log . We shall denote the divisor of marked points on Y by E ⊆ Y . p 106 Let n ≥ 2 be a natural number. Suppose that we have a coherent sheaf with connec- tion (F, ∇ F ) on Y log , where F is killed by p n and ﬂat over Z/p n Z, and the connection ∇ F is relative to the morphism h log : Y log → S log . Suppose, moreover, that we are given a ﬁltration F 1 (F) F p ⊆ F F p of the reduction of F modulo p. We shall call this ﬁltration the Hodge ﬁltration. Then, relative to this data, we deﬁne the coherent sheaf with con- /S log ). nection F ∗ (F, ∇ F ) as follows. First, we regard (F, ∇ F ) as a crystal on Crys(Y F log p Thus, if we apply the relative Frobenius morphism Φ X log /S log to this crystal (F, ∇ F ), we def obtain a crystal (F, ∇ F ) = Φ ∗ X log /S log (F, ∇ F ) on Crys(X F log /S log ). Next, we consider p the subsheaf Φ ∗ X log /S log F 1 (F) F p ⊆ F F p . If we then consider the subsheaf of (F, ∇ F ) on Crys(X F log /S log ) consisting of sections whose reduction modulo p is contained in the p subsheaf Φ ∗ X log /S log F 1 (F) F p , we obtain a crystal (F, ∇ F ) on Crys(X F log /S log ). We then p let F ∗ (E, ∇ E ) = (F, ∇ F ) ⊗ Z p Z/p n−1 Z def Deﬁnition 2.4. We shall call F ∗ (F, ∇ F ) the renormalized Frobenius pull-back of (F, ∇ F ). Note that if F is a vector bundle on Y Z/p n Z , and F 1 (F) F p is a vector bundle on Y F p , with the injection F 1 (F) F p → F F p locally split, it follows immediately from the deﬁnitions that the coherent sheaf F ∗ (F) appearing in F ∗ (F, ∇ F ) is a vector bundle on X Z/p n−1 Z . Note also that, if we think of the “input variable” (F, ∇ F ) as a crystal on Crys(Y F log /S log ), then p . F ∗ (F, ∇ F ) does not depend on the choice of the deformation h log : Y log → S log of Y F log p log log n Often, we will be given a Frobenius lifting Φ log : S Z/p n Z → S Z/p n Z modulo p , and we log log log log will take Y Z/p n Z → S Z/p n Z (respectively, (F, ∇ F )) to be the pull-back of X Z/p n Z → S Z/p n Z (respectively, some (E, ∇ E ) on X log ) by Φ log . If S were the spectrum of the ring of Witt vectors of a perfect ﬁeld k, and n = 2, then the F ∗ (E, ∇ E ) F that we have deﬁned here would coincide with the F ∗ (E, ∇ E ) of Chapter II, Deﬁnition 2.9. log Now let (F, ∇ F ) be a vector bundle with connection on Y Z/p n Z whose determinant n−1 is trivial and which is indigenous modulo p . We will denote its Hodge ﬁltration by F 1 (F) Z/p n−1 Z ⊆ F Z/p n−1 Z . Let us denote by (G, ∇ G ) the vector bundle with connection on X Z/p n−1 Z which is the renormalized Frobenius pull-back of (F, ∇ F ). Suppose, more- ∼ over, that (G, ∇ G ) F F p = (F, ∇ F ) F p . Thus, one sees (as in the proof of Chapter II, Propo- sition 2.10) that (F, ∇ F ) F p is nilpotent and admissible (hence corresponds, by Chapter II, Proposition 2.5, to some FL-bundle). Lemma 2.5. Let n ≥ 3. If we modify the connection ∇ F by some p n−2 θ, where θ is a section of h ∗ (ω Y log /S ) ⊗2 (−E), then the vector bundle F ∗ (F) (on X Z/p n−1 Z ), along with its connection F ∗ (∇ F ), remain unchanged. 107 Proof. Looking at the deﬁnition of the renormalized Frobenius pull-back, one sees that the pair (F ∗ (F), F ∗ (∇ F )) is constructed by pulling back F (and ∇ F ) via various local liftings of Φ X log /S log , and then gluing together by means of gluing morphisms deﬁned by the connection ∇ F . Moreover, these gluing morphisms are obtained from the Taylor expansion (cf. [Falt], §2, Theorem 2.3), which involves applying the connection ∇ F to tangent vectors pushed forward from the Frobenius lifting. Since such tangent vectors are necessarily divisible by p (as well as being annihilated, of course, by p n ), it follows that a knowledge of (∇ F ) Z/p n−1 Z suﬃces to compute these Taylor expansions. Thus, certainly F ∗ (F) depends at most on (∇ F ) Z/p n−1 Z . On the other hand, since at the end of the construction of F ∗ (F), we mod out by p n−1 · Φ ∗ X log /S log F 1 (F) F p ⊆ p n−1 · Φ ∗ X log /S log F F p , we see that modifying ∇ F by an endomorphism-valued diﬀerential whose image lies inside p n−2 · F 1 (F) (where we have p n−2 rather than p n−1 since we always get an extra factor of p from the fact that we are applying the connection to tangent vectors divisible by p) does not aﬀect the result. This completes the proof. Now let us assume that (F, ∇ F ) is a rank two vector bundle on Y Z/p n Z with a con- nection (relative to h log : Y log → S log ), whose determinant is trivial. Let us suppose, moreover, that (F, ∇ F ) Z/p n−1 Z is indigenous. Let (G, ∇ G ) = F ∗ (F, ∇ F ). As before, we ∗ ∼ assume that (G, ∇ G ) F F p = (F, ∇ F ) F p . Then by considering the result of applying F to var- ious deformations (F, ∇ F ) of (F, ∇ F ) (i.e., such that (F, ∇ F ) Z/p n−1 Z = (F, ∇ F ) Z/p n−1 Z ) to obtain various deformations (G, ∇ G ) of (G, ∇ G ), we obtain a morphism: (R 1 f DR,∗ Ad(E) F p ) F → R 1 f DR,∗ Ad(G) F p ∼ = R 1 f DR,∗ Ad(E) F p If we then compose this morphism with the projection R 1 f DR,∗ Ad(E) F p → R 1 f ∗ (τ X log /S log ) F p arising from the Hodge ﬁltration, we obtain a morphism (R 1 f DR,∗ Ad(E) F p ) F → R 1 f ∗ (τ X log /S log ) F p which, by Lemma 2.5, vanishes on the subbundle log ⊗2 1 F ) (−D)) F (f ∗ (ω X/S F p ⊆ (R f DR,∗ Ad(E) F p ) arising from the Hodge ﬁltration. Thus, we obtain a morphism of vector bundles Θ F ∗ : (R 1 f ∗ (τ X log /S log ) F p ) F → R 1 f ∗ (τ X log /S log ) F p Note that by Lemma 2.5, this morphism remains unchanged if one adds some p n−2 θ to the connection ∇ F . 108 Lemma 2.6. The morphism Θ F ∗ is equal to −Φ τ E . Proof. In the gluing process referred to in the proof of Lemma 2.5, the deforming n−1 cocycle in (τ X log /S log ) F (τ X log /S log ) F (mod p n ) only aﬀects the Taylor expan- F p = p sion to ﬁrst order. Moreover, this cocycle in (τ X log /S log ) F F p is mapped to a cocycle in ∗ F Φ X log /S log (τ X log /S log ) F p → Ad(G) F p ∼ = Ad(E) F p , and hence to a cocyle in Ad(E) F p . If we then further project this cocycle via Ad(E) F p → (τ X log /S log ) F p , we obtain Θ F ∗ of the original cocycle. On the other hand, let us note that by Chapter II, Proposition 1.4, the inclusion Φ ∗ X log /S log (τ X log /S log ) F F p → Ad(G) F p = Ad(E) F p is −1 times the p-curvature of τ (E, ∇ E ) F p . Since Φ E is deﬁned by applying R 1 f ∗ to the p-curvature composed with the projection Ad(E) F p → (τ X log /S log ) F p , we thus obtain the result. We are now ready to begin deﬁning a canonical Frobenius lifting on S log , which will be fundamental to the entire paper. First, note that since (N g,r ) F p ⊆ (S g,r ) F p , we have a ord tautological trivialization (τ N ) F p : (N g,r ) ord F p → (S g,r ) F p of the torsor S g,r over (N g,r ) F p . If we pull this trivialization back to S, we get a trivialization (τ S ) F p : S F p → (S g,r ) F p , . This indige- which thus deﬁnes a nilpotent, ordinary indigenous bundle (E, ∇ E ) 1 on X F log p nous bundle thus corresponds to an FL-bundle, hence a section of the torsor D (of the log log previous subsection) over S F p , and hence a Frobenius lifting Φ log 2 : S Z/p 2 Z → S Z/p 2 Z . Now log let (E, ∇ E ) 2 be any indigenous bundle on X Z/p 2 Z that lifts (E, ∇ E ) 1 . (Such a lifting exists log log log since S F p is aﬃne.) We shall deﬁne Y log inductively. Let Y Z/p . 2 Z = X Z/p 2 Z × S log ,Φ log S 2 ∗ ∗ Let (F, ∇ F ) 2 = (Φ log 2 ) (E, ∇ E ) 2 . Then it is a tautology that if we take F (F, ∇ F ) 2 , we obtain (E, ∇ E ) 1 (up to tensor product with a line bundle with connection whose square is trivial; as usual, for the sake of simplicity, we shall ignore this). So far, to summarize, log of the objects constructed so far, Φ log 2 ; Y Z/p 2 Z ; and (E, ∇ E ) 1 are canonical. The primed objects are not canonical. log Let (F, ∇ F ) 3 be any rank two bundle with connection on Y Z/p 3 Z whose determinant is trivial, and whose reduction modulo p 2 is equal to (F, ∇ F ) 2 . That is, (F, ∇ F ) 3 is a deformation of (F, ∇ F ) 2 . Now by Lemma 2.6, and the fact that (E, ∇ E ) 1 is ordinary, it follows that, among all possible deformations (F, ∇ F ) 3 of (F, ∇ F ) 2 , there exists a unique (up to changing the connection by some p 2 · θ) such deformation (F, ∇ F ) 3 such that log log log F ∗ (F, ∇ F ) 3 is indigenous on X Z/p 2 Z . Let Y Z/p 3 Z be the unique deformation of Y Z/p 2 Z log such that when one evaluates the crystal (F, ∇ F ) 3 on Y Z/p 3 Z , it becomes indigenous. ∗ Let (E, ∇ E ) 2 = F (F, ∇ F ) 3 . By Lemma 2.5, (E, ∇ E ) 2 is independent of the choice of log log log ∗ : S Z/p (E, ∇ E ) 2 or (F, ∇ F ) 3 . Let (F, ∇ F ) 2 = (Φ log 3 Z → S Z/p 3 Z be 2 ) (E, ∇ E ) 2 . Let Φ 3 log log the classifying morphism of the r-pointed stable curve of genus g given by Y Z/p 3 Z → S Z/p 3 Z . log log log Thus, Φ log 3 lifts Φ 2 . Again, to summarize, the objects Φ 3 ; Y Z/p 3 Z ; and (E, ∇ E ) 2 (as well log as (F, ∇ F ) 2 ) are canonical. If we now let (E, ∇ E ) 3 be an indigenous bundle on X Z/p 3 Z 109 ∗ that lifts (E, ∇ E ) 2 , and (F, ∇ F ) 3 = (Φ log 3 ) (E, ∇ E ) 3 (where the primed bundles are newly chosen here, hence diﬀerent from the temporary ones we chose before), it follows from Lemma 2.5 that (E, ∇ E ) 2 ∼ = F ∗ (F, ∇ F ) 3 . Continuing in this fashion (making repeated use of Lemmas 2.5 and 2.6, as well as the fact that (E, ∇ E ) 1 is ordinary), we thus obtain a canonical Frobenius lifting Φ log : S log → S log (of p-adic formal schemes), as well as a canonical indigenous bundle (E, ∇ E ) on X log such that F ∗ (Φ log ) ∗ (E, ∇ E ) ∼ = (E, ∇ E ) (up to tensor product with a line bundle with connection whose square is trivial). Moreover, note that by Proposition 2.3, this Frobenius lifting Φ log is ordinary. Deﬁnition 2.7. Let Ψ log : S log → S log be a Frobenius lifting. We shall call an indigenous bundle (G, ∇ G ) on X log Frobenius invariant for Ψ log if (G, ∇ G ) ∼ = F ∗ (Ψ log ) ∗ (G, ∇ G ) (up to tensor product with a line bundle with connection whose square is trivial). So far, we have been working over our aﬃne scheme S, which is étale over M g,r . However, since the objects that we have constructed (namely, Φ log and (E, ∇ E )) are canonical, i.e. uniquely characterized by certain properties that have nothing special to do with S, it is ord clear that they all descend to (N g,r ) log . We thus see that we have proven the following key result: ord Theorem 2.8. On (N g,r ) log , there exists a canonical ordinary Frobenius lifting ord ord log Φ log → (N g,r ) log N : (N g,r ) ord together with a canonical indigenous bundle (E N , ∇ E N ) on C log (where C log → N g,r is the universal r-pointed curve of genus g) whose reduction modulo p is equal to the nilpotent, ordinary indigenous bundle deﬁned by the tautological trivialization (τ N ) F p of S g,r over ord N g,r . Moreover, the pair {Φ log N ; (E N , ∇ E N )} is uniquely characterized by the following properties: ord ord log → (N g,r ) log is a lifting of Frobenius; (1) Φ log N : (N g,r ) (2) the reduction of (E N , ∇ E N ) modulo p is the bundle deﬁned by (τ N ) F p ; (3) (E N , ∇ E N ) is Frobenius invariant for Φ log N . Moreover, the formation of Φ log N and (E N , ∇ E N ) is compatible with restriction to products ord ord of N g,r ’s for smaller g’s and r’s that map into the boundary of our original N g,r via the gluing procedure described at the end of Chapter I, §2. 110 Proof. We have proven everything except the last statement about restriction. To see ord this, note ﬁrst of all that Φ log N respects such products of smaller N g,r since it respects the ord log structure of the original N g,r . Thus, we may restrict Φ log N and (E N , ∇ E N ) to these products, and the result follows by uniqueness. Remark. This result is the central result of this paper. In some sense, the rest of the paper is just devoted to making explicit a number of formal consequences of Theorem 2.8. In particular, since this canonical Frobenius lifting is ordinary, it follows that we can apply the theory of §1. We shall proceed to do this in the remainder of this Chapter. Finally, it is useful to know that the formation of the canonical Frobenius and indige- ord nous bundle are compatible with ﬁnite coverings. Suppose that S log → (N g,r ) log is log étale, with S formally smooth over Z p , and the log structure given by a relative divisor with normal crossings over Z p . Let f log : X log → S log be the pull-back of the universal log curve over M g,r . Let q, s ≥ 0 be such that 2q − 2 + s ≥ 1. Let Y log → S log be an s-pointed stable curve of genus q. Suppose that we are given a morphism over S log : Y log φ log −→ X log Now we make the following: Deﬁnition 2.9. We shall say that φ log is log admissible if it is ﬁnite, log étale, and takes marked points to marked points. A typical example of a log admissible morphism may be obtained by considering the “admissible coverings” of [HM]. Indeed, it is not diﬃcult to see that by endowing the curves involved (as well as the base) with appropriate log structures, one may obtain a log admissible covering (cf. [Mzk], §3). (Note, however, that the deﬁnition of “log admissible” given here diﬀers from that of [Mzk], §3.) Let (E, ∇ E ) be the restriction of the canonical indigenous bundle (E N , ∇ E N ) to X log . log (which exists Let Φ log : S log → S log be the pull-back of the Frobenius lifting Φ log N to S ord because S log → (N g,r ) log is log étale). Let (F, ∇ F ) = φ ∗ (E, ∇ E ). Observe that (F, ∇ F ) F p is a nilpotent, admissible indigenous bundle on Y log . Let us assume that: (*) (F, ∇ F ) F p is ordinary. log Then (F, ∇ F ) F p determines a factorization of the classifying morphism S log → M q,s ord through (N q,s ) log . Thus, we get a morphism ord κ log : S log → (N q,s ) log 111 ord For simplicity, let us write T log for (N q,s ) log . Let us denote by Ψ log the canonical Frobenius ord on (N q,s ) log , and by (G, ∇ G ) the canonical indigenous bundle on the universal s-pointed ord stable curve of genus q over (N q,s ) log . Then we have the following compatibility result: Theorem 2.10. We have a commutative diagram: Φ log S log −→ ⏐ ⏐ log κ T log log Ψ −→ S log ⏐ ⏐ log κ T log and an isomorphism κ ∗ (G, ∇ G ) ∼ = (F, ∇ F ). Proof. We shall apply induction on i to the proposition “the Theorem is true when the objects in it are reduced modulo p i .” The case i = 1 is clear. Thus, it suﬃces to prove the induction step. Let us consider the crystals (F, ∇ F ) Φ and κ ∗ (G, ∇ G ) Ψ on Crys(X F log /S log ). p Suppose that they agree modulo p i . If we apply F ∗ (the renormalized Frobenius pull- back) to them, we get the same crystal modulo p i , by the induction hypothesis and the deﬁnition of the canonical Frobenii and indigenous bundles. Thus, by Lemma 2.6, it follows that the underlying vector bundles of (F, ∇ F ) Φ and κ ∗ (G, ∇ G ) Ψ must agree modulo p i+1 . Since (F, ∇ F ) Φ is indigenous on (Y log ) Φ , and κ ∗ (G, ∇ G ) Ψ is indigenous on the s-pointed stable curve of genus q given by pulling back the universal one by Ψ log ◦ κ log , we thus obtain that the diagram in the Theorem commutes modulo p i+1 . Then since (F, ∇ F ) and κ ∗ (G, ∇ G ) agree modulo p i , it follows that their underlying vector bundles agree modulo p i+1 . By a similar argument, their underlying vector bundles also agree modulo p i+2 , and the diagram commutes modulo p i+2 . Then, by Lemma 2.5, since (F, ∇ F ) ∼ = F ∗ (F, ∇ F ) Φ ∗ ∗ ∗ Ψ ∗ ∼ and κ (G, ∇ G ) = κ F (G, ∇ G ) , it follows that (F, ∇ F ) and κ (G, ∇ G ) agree modulo p i+1 . This completes the proof of the induction step. §3. Applications of the Canonical Frobenius Lifting In this Section, we apply the general theory of §1 to the canonical modular Frobenius lifting constructed in §2. In particular, we deﬁne the notion of a p-adic quasiconformal equivalence class, and show how the choice of such a class allows one to construct both aﬃne and multiplicative uniformizations of M g,r . We will also deﬁne the notion of a p- adic topological marking, which will allow us to construct a local uniformization of M g,r by means of the aﬃne space of quadratic diﬀerentials. As we make these constructions, we will compare them to various classical constructions in the complex case. Finally, we will specialize what we have done in this Chapter to the case of elliptic curves (regarded 112 parabolically) to see that in this case, the canonical Frobenius lifting corresponds to a well-known Frobenius lifting from Serre-Tate theory, and that, consequently, the various objects constructed from it – i.e., canonical curves, modular uniformizations, etc. – reduce to the corresponding objects of classical Serre-Tate theory. Canonical Liftings of Curves over Witt Vectors ord Let N g,r ; Φ log N be as in the last subsection of §2. Let k be a perfect ﬁeld of characteristic p. Let A = W (k), the ring of Witt vectors with coeﬃcients in k; let S = Spec(A). Thus, we have a natural Frobenius automorphism Φ A : A → A on A. Recall the notion of a ord canonical liftings of A-valued points in N g,r (Deﬁnition 1.9). ord Deﬁnition 3.1. We shall call a point α 0 ∈ N g,r (k) a (p-adic) quasiconformal equivalence class (valued in k). We shall call an r-pointed stable curve of genus g the canonical curve in the class α 0 if it admits an indigenous bundle such that the pair consisting of the curve ord and this indigenous bundle deﬁnes a canonical A-valued point of N g,r whose reduction modulo p is α 0 . Remark. Thus, a p-adic quasiconformal equivalence class consists of a hyperbolically ordi- nary r-pointed stable curve (X 0 → Spec(k); p 1 , . . . , p r : Spec(k) → X 0 ) of genus g, together with a choice of a nilpotent, ordinary indigenous bundle (E, ∇ E ) 0 on X 0 log . Recall from Chapter II, Proposition 3.13, that for a given ordinary X 0 log , there are at most p 3g−3+r possible choices for (E, ∇ E ) 0 . The reason for attaching the term “quasiconformal” to this data will become more and more apparent as we continue: Namely, unlike the complex case in which, once g and r are determined, all curves belong to the same quasiconformal equiv- alence class, the uniformization theory that we shall develop in this paper in the p-adic setting acts (by comparison to the classical complex case) as if there are many diﬀerent quasiconformal equivalence classes (for a given g and r), and moreover, this equivalence ord class is determined exactly by the datum of a point in N g,r . Specializing the theory of §1, we obtain: ord Theorem 3.2. For every p-adic quasiconformal equivalence class α 0 ∈ N g,r (k), there ord exists a canonical lifting α ∞ ∈ N g,r (A), i.e., more concretely, an r-pointed stable curve (X → Spec(A); p 1 , . . . , p r : Spec(A) → X) of genus g, together with an indigenous (E, ∇ E ) on X log . This canonical lifting α ∞ is uniquely characterized by the fact that it is ﬁxed log log under Φ −1 A Φ N , where Φ N is the canonical Frobenius lifting of Theorem 2.8. Corollary 3.3. Suppose that the pair 113 {(X → Spec(A); p 1 , . . . , p r : Spec(A) → X); (E, ∇ E )} is canonical (i.e., for (E, ∇ E ), this means that it is the restriction of the (E N , ∇ E N ) of Theorem 2.8). Then (1) If X → Spec(A) is smooth, then (E, ∇ E ) deﬁnes an MF ∇ -object on X log in the sense of [Falt], §2 (up to tensor product with a line bundle whose square is trivial). (2) More generally, if X → Spec(A) is not smooth, then the pair is obtained by gluing together (as at the end of Chapter I, §2) a collection of smooth canonical pairs. Proof. The two statements follow by specializing Theorem 2.8. Corollary 3.4. A pair {(X → Spec(A); p 1 , . . . , p r : Spec(A) → X); (E, ∇ E )} consisting of a smooth r-pointed curve of genus g and an indigenous bundle on X log is canonical if and only if (1) the indigenous bundle (E, ∇ E ) F p is ordinary; (2) (E, ∇ E ) deﬁnes an MF ∇ -object on X log in the sense of [Falt], §2 (up to tensor product with a line bundle whose square is trivial). More generally, a pair consisting of an r-pointed stable curve of genus g and an indigenous bundle on X log is canonical if and only if it is obtained by gluing together canonical pairs as at the end of Chapter I, §2. Proof. Let us ﬁrst consider the smooth case. By the previous Corollary, it suﬃces to prove the “if” part. Since (E, ∇ E ) is an MF ∇ -object, we know (by Chapter II, Proposition 2.10) that (E, ∇ E ) F p is nilpotent. Thus, there exists a canonical pair which is equal to our given pair modulo p. On the other hand, it follows by the analogues of Lemmas 2.5 and 2.6 for F ∗ over S = Spec(A) that there is only one lifting of our pair modulo p that admits an indigenous bundle which is an MF ∇ -object. Thus, our pair must be the canonical pair. Next, we consider the stable case. Again, by the previous Corollary, it suﬃces to prove the “if” part. As before, we note that there exists a canonical pair which is equal to our given pair modulo p. By the previous Corollary, the canonical pair is obtained by gluing together smooth canonical pairs. Since a smooth canonical lifting is unique, it thus follows that our pair must be the canonical pair. 114 Corollary 3.5. Suppose that we have an r-pointed (respectively, s-pointed) stable curve X log (respectively, Y log ) of genus g (respectively, q) over S log (for some appropriate choice of log structure on S log ). Let (E, ∇ E ) (respectively, (F, ∇ F )) be an indigenous bundle on X log (respectively, Y log ). Suppose that (F, ∇ F ) F p is ordinary. Let φ log : Y log → X log be log admissible, and suppose that (F, ∇ F ) = φ ∗ (E, ∇ E ). Then the pair {X log ; (E, ∇ E )} is canonical if and only if the pair {Y log ; (F, ∇ F )} is canonical. ω Proof. First note that, by pulling back square diﬀerentials and considering Φ ω E and Φ F , the ordinariness of (F, ∇ F ) F p implies the ordinariness of (E, ∇ E ) F p . The stipulated conditions on φ log imply that X log is obtained by gluing if and only if Y log is obtained by gluing. We thus reduce to the smooth case. But this follows immediately, by the criterion of the previous Corollary (about the indigenous bundle being an MF ∇ -object) and the fact that F ∗ commutes with log étale coverings. Canonical Aﬃne Coordinates on M g,r We maintain the notation of the preceding subsection. Thus, S = Spec(A); A = W (k); ord and k is a perfect ﬁeld of characteristic p. Let α ∈ N g,r (A) be canonical. Since giving ord a canonical α ∈ N g,r (A) is equivalent to giving the p-adic quasiconformal equivalence ord class α(mod p) ∈ N g,r (k), we shall frequently abuse notation and speak of “the p-adic quasiconformal equivalence class α.” Let us assume that α corresponds to a smooth curve. Then applying Theorems 1.12 and 1.14 to the canonical Frobenius lifting of Theorem 2.8 gives the following results: Theorem 3.6. For every choice of a p-adic quasiconformal equivalence class α, we obtain a local canonical uniformization ∼ α U can : Θ aﬀ α = M g,r of M g,r by the aﬃne space modeled on Θ α . Let (B, m B ) be a local ring with residue ﬁeld k which is p-adically complete and has a topologically nilpotent PD-structure on m B . ord Deﬁnition 3.7. We shall say that β ∈ N g,r (B) is in the (p-adic) quasiconformal equiva- ord ord lence class α if the point β(mod m B ) ∈ N g,r (k) is equal to α(mod p) ∈ N g,r (k). In summary, we have proven the following result: 115 ord Theorem 3.8. For every β ∈ N g,r (B) in the quasiconformal equivalence class α, we obtain a canonical class κ β ∈ m B · Θ β , as well as a local uniformization ∼ β U β : Θ aﬀ β = M g,r of M g,r by the aﬃne space modeled on Θ β . Moreover, this uniformization is related to the canonical uniformization by tensoring over A with B, applying the isomorphism −1 : (Θ α ) B → Θ β (Ψ ∨ αβ ) ord and then translating by κ β . Finally, for all β ∈ N g,r (B) that lie in the quasiconformal equivalence class α, the correspondence β → κ β is a bijection of such β onto m B · Θ β . Remark. In considering the uniformizations just obtained, it is tempting to compare them with the local uniformization by the aﬃne space modeled on the tangent space to M g,r given in the complex case by considering geodesics for the Teichmüller metric. We believe, however, that if there is any proper complex analogue to the uniformizations of Theorems 3.6 and 3.8 at all, then it is the uniformization obtained by Bers coordinates. Indeed, unlike the Teichmüller coordinates, which are real, but not complex analytic, the Bers coordinates are (complex) analytic, just as the aﬃne coordinates of Theorems 3.6 and 3.8. Also, (perhaps more crucially) the Teichmüller coordinates are the same for elliptic curves regarded either hyperbolically or parabolically. We shall soon see, however, that the uniformizations analogous to those of Theorems 3.6 and 3.8 for elliptic curves treated parabolically are diﬀerent from those in the hyperbolic case. One diﬀerence between the Bers uniformization and the uniformizations of Theorems 3.6 and 3.8 is that the Bers uniformization is by the aﬃne space of quadratic diﬀerentials (of the complex conjugate curve), not by the tangent space to M g,r . On the other hand, (even in the complex case) one cannot have a holomorphic local canonical uniformization by the aﬃne space of quadratic diﬀerentials, as one can see easily by considering a one-pointed curve of genus one with an automorphism of degree three. Thus, to obtain a uniformization by quadratic diﬀerentials, we need more “rigidifying” information. In our case, the information will take the form of a topological marking of the curve. Topological Markings and Uniformization by Quadratic Diﬀerentials We maintain the notation of the previous subsection. Let us consider the canonical ord ord Frobenius lifting Φ N : N g,r → N g,r of Theorem 2.8. Just as in Deﬁnition 1.2, Φ N deﬁnes ord canonical étale local systems on N g,r in free Z p -modules of rank 3g − 3 + r: ord Deﬁnition 3.9. We shall refer to Θ et N as the tangential local system on N g,r . We shall ord call its dual, Ω et N , the diﬀerential local system on N g,r . 116 Note that if one tensors over Z p with F p , then these local systems become the local systems (with the same names) considered in Chapter II, §3, following Proposition 3.4. ord Now let us assume that k is algebraically closed. Let α ∈ N g,r (A) be a p-adic quasi- conformal equivalence class. In this subsection, however, we assume that α corresponds to ord a smooth curve. We would like to consider the fundamental groupoid of N g,r in the sense of [SGA 1], Exposé V, p. 130. Recall that this fundamental groupoid is the category of ord ﬁber functors from the category of ﬁnite étale coverings of N g,r to the category of sets. ord Moreover, if x ∈ N g,r (k), then x deﬁnes a ﬁber functor F x of this type by simply restrict- ord ord ing étale coverings of N g,r to Spec(k) via pull-back by x. Thus, if x, y ∈ N g,r (k), then we shall call a path from x to y a natural transformation from F x to F y . On the other hand, recall from the last subsection of Chapter II, §3, the set D g,r of isomorphism classes of totally degenerate r-pointed stable curves of genus g. If Δ ∈ D g,r , ord then, by abuse of notation, we shall also write Δ ∈ N g,r (A) for the totally degenerate curve over A represented by Δ. Now we make the following important Deﬁnition 3.10. We shall call a pair μ = (Δ; ) a (p-adic) topological marking for the ord quasiconformal equivalence class α if Δ ∈ D g,r and is a path from α(mod p) ∈ N g,r (k) ord to Δ ∈ N g,r (k). Let μ = (Δ; ) be a topological marking for α. Then let us note that μ deﬁnes a canonical nondegenerate bilinear form B μ on Θ α as follows. First note that by the construction in Proposition 3.6 of Chapter II, §3 (which is carried out there over F p , but et clearly works just as well over Z p ), we have a canonical nondegenerate bilinear form B Δ et ∼ et on Θ et Δ . Now the path deﬁnes an isomorphism Θ : Θ α = Θ Δ . Thus, if we pull-back et by means of Θ , we get a canonical nondegenerate bilinear form B μ et on Θ et B Δ α . Since Θ α = (Θ et ) ⊗ A, we thus obtain (by tensoring) a canonical nondegenerate bilinear form Z p α B μ on Θ α . Now let (B, m B ) be a local ring with residue ﬁeld k which is p-adically complete and has ord a topologically nilpotent PD-structure on m B . Let β ∈ N g,r (B) be in the quasiconformal equivalence class α. Recall the canonical isomorphism −1 : (Θ α ) B → Θ β (Ψ ∨ αβ ) implicit in Theorem 3.8. This isomorphism allows us to transport B μ to Θ β so as to obtain a canonical nondegenerate bilinear form B μ β on Θ β . We summarize this as follows: Proposition 3.11. The choice of a topological marking μ on a quasiconformal equivalence class α allows one to deﬁne a canonical nondegenerate bilinear form B μ β on Θ β for every ord β ∈ N g,r (B) in the quasiconformal equivalence class α. 117 This ﬁnally allows us to give local uniformizations of M g,r by means of quadratic diﬀeren- tials: Namely, we compose the aﬃne uniformization of Theorem 3.8 with the isomorphism Θ β ∼ = Ω β given by the nondegenerate bilinear form B μ β : Theorem 3.12. The choice of a topological marking μ on a quasiconformal equivalence class α that corresponds to a smooth curve allows one to deﬁne a canonical class κ ∨ μ,β ∈ m B · Ω β , as well as a local uniformization ∼ β V μ,β : Ω aﬀ β = M g,r ord of M g,r by the aﬃne space modeled on Ω β , for every β ∈ N g,r (B) in the quasiconformal ord equivalence class α. Finally, for all β ∈ N g,r (B) that lie in the quasiconformal equivalence class α, the correspondence β → κ μ,β is a bijection of such β onto m B · Ω β . Remark. Thus, we have obtained a canonical uniformization of M g,r by quadratic diﬀer- entials for every choice of a topological marking on α. In the complex case, a topological marking of a Riemann surface is given by ﬁxing the underlying topological manifold, up to homeomorphisms homotopic to the identity. Thus, the analogy between topological markings in the p-adic and complex cases lies in the fact that a p-adic topological marking gives one a canonical basis for Θ et β , hence for Θ β , corresponding to a collection of parti- tion curves (see Introduction, §2) of a Riemann surface. This speciﬁcation of partition curves determines a topological marking, by gluing together “pants” along the partition curves. Thus, instead of uniformizing by the aﬃne space modeled on Ω β , we could also have uniformized by the aﬃne space modeled on a direct product of aﬃne lines, one for each “partition curve.” Whichever choice of coordinates (i.e., quadratic diﬀerentials or partition curves) is more useful depends on one’s tastes or the applications one has in mind. Canonical Multiplicative Parameters So far we have only been working with smooth curves. In order to ﬁnd canonical parameters at singular curves, we need to work with multiplicative parameters (like the q-parameter in the case of elliptic curves), as opposed to aﬃne parameters, as in Theorems 3.6 and 3.8. ord Let α ∈ N g,r (A) be a p-adic quasiconformal equivalence class (corresponding to a curve which is not necessarily smooth). Let us assume, for the rest of this subsection, that k is algebraically closed. Let Ω log α be the restriction of Ω M log to α, and let Θ α be g,r the dual A-module to Ω log α . Then the Frobenius invariant subsections of Θ α form a free et log Z p -submodule Θ et ⊆ Θ α of rank 3g − 3 + r. Similarly, we have Ω α ⊆ Ω α . Let (M g,r ) α be α the completion of M g,r ⊗ Z p A at the image of α. Let ω ∈ Ω et α have residues equal to zero 118 or one at all the irreducible components of the divisor at inﬁnity of M g,r , and nonzero reduction modulo p. Then, just as in Deﬁnition 1.11, we have a parameter q ω,α on (M g,r ) α , which is well-deﬁned up to multiplication by a Teichmüller representative [k × ]. This parameter is a unit at all the divisors where the residue of ω is zero and has valuation one at all the divisors where the residue of ω is one. Moreover, p Φ −1 N (q ω,z ) = q ω,z Deﬁnition 3.13. We shall call such a parameter q ω,α a canonical multiplicative parameter on (M g,r ) α . The Case of Elliptic Curves Just as in previous Chapters, it is useful to look at the case of elliptic curves (regarded parabolically) since the calculations are usually much easier in this case. As before, we let log log M 1,0 be the log stack of elliptic curves, and f log : G log → M 1,0 be the universal elliptic curve (with the log structure deﬁned by the pull-back to G of the divisor at inﬁnity of ord M 1,0 ). Let M 1,0 ⊆ M 1,0 be the open p-adic formal substack parametrizing ordinary ord ord elliptic curves. Recall that we computed in Chapter II, Theorem 3.11, that N 1,0 = M 1,0 , ord and that the section of S 1,0 over (M 1,0 ) F p corresponding to the unique nilpotent, ordinary indigenous bundle on an elliptic curve was given explicitly in Example 2 of Chapter I, §2. Now it is easy to see that, although nominally everything in this Chapter was done for hyperbolic curves, much of the theory goes through for elliptic curves, as well. In ord particular, the construction of the canonical Frobenius lifting Φ N on M 1,0 goes through just as before. Since everything else in the Chapter is essentially a formal consequence of the existence of Φ N , in this subsection, we would like to compute the lifting Φ N explicitly for elliptic curves, and identify the resulting concepts (i.e., canonical curves, uniformization, topological marking, etc.) with the well-known objects of classical Serre-Tate theory. For a treatment of classical Serre-Tate theory, we refer to [Mess] and [KM] (p. 260). ord Let us begin by recalling a certain Frobenius lifting Φ M on M 1,0 which is fundamental to Serre-Tate theory. Ultimately, we shall show that Φ M = Φ N . First recall that the étale quotient of the (log) p-divisible group P associated to the universal elliptic curve ord ord G ord → M 1,0 deﬁnes a local system L on M 1,0 in free Z p -modules of rank one. Also, since P is self-dual, taking Cartier duals gives us an inclusion L ∨ (1) ⊗ (Q p /Z p ) → P (where the “1” in parentheses denotes a Tate twist). Let P Φ ⊆ P be the subgroup scheme given by L ∨ (1) ⊗ ( p 1 Z p /Z p ). Thus, taking the quotient by this subgroup scheme P Φ gives us an isogeny: 119 Φ G : G ord → H ord ord ord to some elliptic curve H over M 1,0 . Let Φ M : M 1,0 → M 1,0 be the classifying morphism of H. Thus, H = Φ ∗M G ord . Since considered modulo p, the subgroup scheme P Φ is nothing but the kernel of Frobenius, it follows that Φ M is a Frobenius lifting, and that (Φ G ) F p is ord just the relative Frobenius on G F . For convenience, we shall denote objects pulled back p via Φ M with a superscript “F .” Now let us consider the eﬀect of pulling back the indigenous bundle (E, ∇ E ) F on (G ) , where (E, ∇ E ) is the indigenous bundle on G ord given in Example 2 of Chapter I, §2. Let (F, ∇ F ) = Φ ∗G (E, ∇ E ) F . Let us denote by ω the relative dualizing sheaf of ord F ord G ord → M 1,0 . Then as a vector bundle, F = Φ ∗G (ω) F ⊕ O G ∗ F Now let Φ ω G : Φ G (ω) → ω denote the morphism on diﬀerentials induced by Φ G , divided by p. Then I claim that Φ ω G is an isomorphism. Indeed, since we are dealing with ordinary elliptic curves, the local group structure near the origin is isomorphic to that of G m (the multiplicative group scheme), and the Frobenius lifting Φ G just amounts to the p th power map on G m . This proves the claim. Since E = ω ⊕ O G , Φ ω G thus gives us an isomorphism: E∼ = F by taking the direct sum of Φ ω G with the identity on O G . Next, we consider connections. Recall that ∇ E diﬀers from the trivial connection by the tautological Ad(E)-valued dif- ferential form given by mapping the ﬁrst factor ω to the second factor O G ⊗ ω. Thus, when we pull-back by Φ G , we get a similar nilpotent endomorphism-valued diﬀerential form, this time given by the map from Φ ∗G (ω) F (the ﬁrst factor) to O G ⊗ ω (the second factor) given by p · Φ ω G . On the other hand, when we compute the renormalized Frobenius pull-back of (E, ∇ E ), we divide out by this factor of p. It thus follows that under the isomor- phism E ∼ = F considered above, the renormalized Frobenius pull-back gives a connection on F which corresponds precisely to the connection ∇ E on E. Since Φ N and (E N , ∇ E N ) are uniquely characterized by the property that the renormalized Frobenius pull-back of Φ ∗N (E N , ∇ E N ) is isomorphic to (E N , ∇ E N ), we thus see that we have proven the following result. Theorem 3.14. The canonical Frobenius lifting Φ N for elliptic curves (regarded parabol- ically) is equal to the Frobenius lifting Φ M . Moreover, the canonical indigenous bundle (E N , ∇ E N ) is the indigenous bundle constructed in Example 2 in Chapter I, §2. Remark. In other words, what we have constructed here is just a relative version of the uniformizing MF ∇ -object of Deﬁnition 1.3. 120 Now let k be a perfect ﬁeld of characteristic p. Then it is well-known from Serre-Tate theory that an elliptic curve E → Spec(W (k)) is canonical in the sense of Serre-Tate the- ory if and only if the point in α ∈ M 1,0 (W (k)) that it deﬁnes is ﬁxed by Φ M . We thus obtain that the deﬁnition of a canonical curve given in Deﬁnition 3.1 is consistent with the deﬁnition arising from Serre-Tate theory. Suppose we ﬁx a trivialization of L ⊗2 | α . Then Serre-Tate theory gives a local uniformization of M 1,0 near this point α by the comple- tion G m of the multiplicative group at the identity. Relative to this uniformization, Φ M becomes the p th power map on G m . It thus follows immediately that the canonical aﬃne parameters that we constructed before (in the general case) correspond to the logarithm of the Serre-Tate parameter (up to multiplication by a unit of W (k)). Moreover, one sees ⊗2 . easily that the local system Ω et M corresponding to the Frobenius lifting Φ M is simply L Thus, a topological marking (in the sense of Deﬁnition 3.10) deﬁnes a trivialization of L ⊗2 | α , and so the Serre-Tate parameter itself is a canonical multiplicative parameter in the sense of Deﬁnition 3.13. We summarize this as follows: Theorem 3.15. Canonical liftings for elliptic curves (as deﬁned in Deﬁnition 3.1 relative to Φ N ) are the same as canonical liftings in the sense of Serre-Tate theory. Moreover, the uniformization of Theorem 3.12 in the case of elliptic curves (regarded parabolically) is the same as the uniformization of M ord 1,0 given by Serre-Tate theory. Remark. It appears that the case discussed here in Theorem 3.15, i.e., the case g = 1, r = 0, is the only case of the theory of this paper that is essentially a reformulation of a classically known theory. For instance, already in the case g = 1, r = 1, despite the fact that ord ord and N 1,1 are quite diﬀerent. M 1,0 = M 1,1 (as stacks), it is not diﬃcult to show that N 1,0 ord Indeed, in general, there exist connected components of N 1,1 that are of degree > 1 over M 1,1 (cf. Proposition 3.12 of Chapter II). This implies, in particular, that Φ N in the case g = 1, r = 1 is quite diﬀerent from Φ N in the case g = 1, r = 0. Finally, we observe that the term “topological marking” is apt in this case in the sense that a topological marking deﬁnes a trivialization of L| α , which is analogous in the complex case to specifying a particular pair of generators for the fundamental group of an elliptic curve. 121 Chapter IV: Canonical Curves §0. Introduction Because canonical curves (as deﬁned in Chapter III, Deﬁnition 3.1) admit Frobenius invariant indigenous bundles, they possess a number of special arithmetic and geometric properties. In this Chapter, we study a number of these properties, foremost among which are the existence of a canonical Frobenius lifting, and the construction of a canonical p- divisible group. In particular, the canonical Frobenius lifting allows us to give a geometric characterization of canonical curves which may be regarded as the hyperbolic analogue of the statement in Serre-Tate theory that a lifting of an ordinary elliptic curve is canonical if and only if it admits a lifting of Frobenius. From the point of view of comparison with the complex case, this canonical Frobenius lifting may be regarded as a sort of p-adic Green’s function. In the complex case, the Green’s function plays a central role in the development of uniformization theory from the classical (as opposed to Bers’ quasiconformal) point of view. In this context, the Green’s function is essentially the logarithm of the hyperbolic distance function between two points. We shall see that the Frobenius lifting also gives us a sort of p-adic notion of distance. Also, we shall see that we can construct “pseudo-Hecke correspondences” which in some sense geometrically codify this notion of distance. On the other hand, in the canonical case, we can also construct a certain Galois rep- resentation (arising from the torsion points of the canonical log p-divisible group) which is the p-adic analogue of the canonical representation in the complex case of the fundamental group into PSL 2 (R) (deﬁned by the covering transformations of the upper half plane). Thus, in some sense, we see that at least in the canonical case, we are able to obtain ana- logues of most of the fundamental objects that appear in classical complex uniformization theory. This brings us to the ﬁnal reason for wanting to study the canonical case: namely, the fact that the universal hyperbolically ordinary curve (over the moduli stack) is itself (essentially) a canonical curve. Thus, in Chapter V, by restricting these canonical objects over the universal curve to a given (not necessarily canonical) curve, we will be able to obtain Green’s functions, canonical Galois representations, and so on for noncanonical curves, as well. §1. The Canonical Galois Representation In this Section, we construct a certain canonical Galois representation of the arith- metic fundamental group of a canonical curve. After studying some of the basic global properties of such representations, we then consider what happens on the ordinary locus of the curve. In particular, we construct a canonical ordinary Frobenius lifting over the ordinary locus. This allows us to apply the general theory of Chapter III, §1. We will 122 refer to the multiplicative parameters obtained from this general theory as the Serre-Tate parameters. We will make use of the Serre-Tate parameter quite often in this Chapter. Throughout this Section, we will work over A = W (k), where k is a perfect ﬁeld of odd characteristic. The quotient ﬁeld of A will be denoted by K. Let g, r be nonnegative ord integers such that 2g − 2 + r ≥ 1. Also, we will deal with a ﬁxed α ∈ N g,r (A), which corresponds to a smooth canonical curve f log : X log → S log , where S log is Spec(A) with the trivial log structure. Since singular canonical curves are just obtained by gluing together smooth canonical curves, we shall concentrate mainly on the smooth case. Construction and Global Properties Let (E, ∇ E ) be the canonical indigenous bundle on X log (whose existence is stated in Chapter III, Theorem 2.8). In fact, unless the number r of marked points is even, such a vector bundle will not exist. However, one can always pass to an étale double cover of X on which it will exist, and then descend. For simplicity, we will just act as though this problem does not exist, except when we state ﬁnal results in Theorems, in which case our representations will be into GL ± (that is, the general linear group GL modulo the subgroup {±1}). Now, we would also like to say that the renormalized Frobenius pull-back F ∗ (E, ∇ E ) F is isomorphic to (E, ∇ E ). In general, this may only be true up to tensoring with a line bundle with connection whose square is trivial, but this may also be ignored, provided we remember that ultimately our representations will be into GL ± , not GL. Let us choose an isomorphism Φ E : (E, ∇ E ) ∼ = F ∗ (E, ∇ E ) F which is the identity on determinants. We shall call Φ E the canonical Frobenius action on (E, ∇ E ). Now let us assume that there exists a rational point x : S → X on X which avoids log the marked points. Let us denote by Π the proﬁnite group π 1 (X K , x K ). Then Theorem 2.6 of [Falt] implies that Theorem 1.1. There exists a unique dual crystalline (in the sense of [Falt], §2) repre- sentation ρ : Π → GL ± (V ) (where V is a free Z p -module of rank two) that corresponds (under the functor D(−) of [Falt], §2) to (E, ∇ E , Φ E ). Moreover, the determinant representation of ρ is the cyclotomic character. We shall refer to ρ as the canonical crystalline representation associated to X log . Remark. In the complex case, a hyperbolic Riemann surface can be uniformized by the upper half plane. Then the fundamental group of the Riemann surface acts on the upper half plane via covering transformations, and so we get a representation of the fundamental 123 group into PSL 2 (R), which is canonically determined up to conjugation. The representa- tion ρ of Theorem 1.1 is the p-adic analogue of this complex representation. def Now let Δ = π 1 (X K , x K ) be the geometric subgroup of Π, so Γ = Π/Δ is the Galois group of K over K. Then by “the comparison theorem” (Theorem 5.3 of [Falt]), we get: Theorem 1.2. Let p ≥ 5. Then the group cohomology modules H i (Δ, Ad(V )(1)) (where the “(1)” is a Tate twist) are zero, except when i = 1. Let U = H 1 (Δ, Ad(V )(1)). Then U is a crystalline Z p -Γ-module, which, as a Z p -module is free of rank 6(g − 1) + 2r. It corresponds under the functor D(−) to an MF-object (in the sense of Fontaine-Laﬀaille) (N ; F i (N ); φ i ) over A, where N is a free A-module of rank 6(g − 1) + 2r; F i (N ) = 0 if log ⊗2 i ≥ 4; F i (N ) = N if i ≤ 0; F i (N ) is naturally isomorphic to H 0 (X, (ω X/S ) (−D)), if 1 1 i = 1, 2, 3; and N/F (N ) is naturally isomorphic to H (X, τ X log /S log ). Remark. Some mathematicians have raised questions concerning that the validity of the proof in [Falt], Theorem 5.3. However, in this one-dimensional case, one can give ad hoc proofs of this result, and, moreover, (at least in the closed case, when r = 0) T. Tsuji has orally informed the author that he has obtained a diﬀerent proof of Theorem 5.3 of [Falt]. One interesting fact about the canonical representation ρ is that it is possible to characterize it – as well as the canonicality of X log – solely in terms of the properties of ρ as a Galois representation: Theorem 1.3. Suppose that p ≥ 5. Let X log → Spec(A) be any (not necessarily canonical) r-pointed smooth curve of genus g over A. Assume that we are not in the cases (g = 0; r = 3) or (g = 1; r = 1). Let τ : Π → GL ± (W ) be any dual crystalline representation of log Π = π 1 (X K , x K ) on a free Z p -module W of rank two such that (1) H P i (Δ, Ad(W )(1)) = 0 if i = 1; H 1 (Δ, Ad(W )(1)) is crystalline, and corresponds to an MF-object M = (M ; F i (M ); ψ i ) such that F i (M ) = 0 if i ≥ 4; F i (M ) = M if i ≤ 0; and F i (M ) is a free A-module of rank 3(g − 1) + r if i = 1, 2, 3; (2) the Frobenius endomorphism of (M/F 1 (M )) F p (arising from the MF- object of (1)) is an isomorphism; (3) det(τ ) is the cyclotomic character. Then X log is canonical, and τ is isomorphic to the representation ρ of Theorem 1.1. Proof. Since τ is asserted to be dual crystalline, it corresponds to some vector bundle with connection (G, ∇ G ) on X log , together with a ﬁltration F i (G) on G. Let i 1 (respectively, i 2 ) be the largest i such that F i (G) = 0 (respectively, F i (G) = G). Thus, i 1 ≥ i 2 . The 124 condition that det(τ ) be cyclotomic implies that i 1 + i 2 = 1. If the rank of F i 1 (G) is not one, then i 1 = i 2 , and det(τ ) could not be cyclotomic, so F i 1 (G) must be of rank one, and i 1 > i 2 . Let L = F i 1 (G). Thus, L is a line bundle. Let j 1 be the largest j such that F j (M ) = 0. Now we claim that L can not be stable under ∇ G . Indeed, if it were, then the monodromy at the marked points of ∇ G on L, being nilpotent and one-dimensional, must be zero. Thus, the induced connection on L has no poles at the marked points. But this would imply that deg(L) = 0. Hence the rank over A of F j 1 (M ) would be ≤ h 0 (X, L ⊗2 ⊗ ω X/S ) ≤ g < 3g − 3 + r (by Cliﬀord’s Theorem), which contradicts our hypotheses. This proves the claim. On the other hand, by Griﬃths transversality, if i 1 − i 2 ≥ 2, then F i 1 (G) would have to be stable under ∇ G . Thus, i 1 = i 2 + 1, so i 1 + i 2 = 1 implies that i 1 = 1 and i 2 = 0. Now rank A (F 3 (M )) = 3g − 3 + r ≤ h 0 (X, L ⊗2 ⊗ ω X/S ), so the line bundle L ⊗2 ⊗ ω X/S must be nonspecial, by Cliﬀord’s Theorem. It thus follows that deg(L ⊗2 ) ≥ 2g−2+r. Since the Kodaira-Spencer morphism for the ﬁltration is nonzero, we cannot have deg(L ⊗2 ) > 2g − 2 + r. Thus, we see that (G, ∇ G ) must be indigenous. Since it is also carried to itself by the renormalized Frobenius, it follows from Chapter III, Corollary 3.4, that X log is canonical, and that (G, ∇ G ) must be the canonical indigenous bundle of Theorem 1.1. Remark. For the reader who is interested in handling the cases g = 0; r = 3 and g = 1; r = 1, as well, we remark that by considering conditions (similar to those imposed on H i (Δ, Ad(W )(1))) on higher symmetric powers of W , one can characterize the canonical representations in these cases as well solely in terms of their properties as Galois represen- tations. Remark. Really, the substantive missing element here is that it is not clear to the author how to characterize the property of being “dual crystalline” solely in terms of proper- ties of the representation relative to the triple (Π; Δ ⊆ Π; Π/Δ ∼ = Gal(K/K)). Thus, log is always present in the background of this ultimately, a knowledge of the curve X “Galois representation-theoretic” characterization of the canonical representation. For in- stance, if the property of being “dual crystalline” were known to depend only on the triple (Π; Δ ⊆ Π; Π/Δ ∼ = Gal(K/K)), then one could obtain the result that whether or not a curve is canonical depends only on that triple. In the following, we return to the assumption that X log is canonical. The Horizontal Section over the Ordinary Locus We maintain the notation of the previous subsection. Let X ord be the p-adic formal scheme which is the open sub-formal scheme of X given by the complement of the super- singular divisor (Chapter II, Proposition 2.6). Let us endow X ord with the log structure induced by X log , and call the resulting log formal scheme (X log ) ord . We shall refer to 125 (X log ) ord as the ordinary locus of X log . The purpose of this subsection is to prove and interpret the following result: 1 Theorem 1.4. There exists a unique subbundle T 2 ⊆ E| X ord of rank one with the following properties: 1 (1) T 2 is horizontal, and moreover, for any n, the reduced line bundle 1 2 T Z/p n Z has a nonempty subsheaf (in the category of sets) consisting of 1 2 horizontal sections that generate T Z/p n Z as an O X ord -module; 1 (2) T 2 is taken to itself by Φ E . Finally, (T 2 ) ⊗2 is naturally isomorphic to τ X log /S log | X ord . 1 1 Proof. Let us prove that there exists a unique Φ E -invariant horizontal subbundle T 2 ⊆ E| X ord with horizontal generating sections. We prove this by induction on n, where the n th 1 step is the construction of such a T 2 modulo p n . For n = 1, recall that (up to tensoring with a line bundle) E F p is an FL-bundle (Chapter II, Proposition 2.5). Then under the 1 correspondence of that Proposition, we take our subbundle T F 2 p to be the subbundle of E corresponding to the subbundle that we called “T ” in our discussion of FL-bundles in Chapter II, §1. This subbundle is clearly horizontal, and has local generating sections that are horizontal. In this case, uniqueness follows from the fact that the p-curvature is nonzero. Now we assume that n ≥ 2, and that the result is known for n−1. Let U log = (X log ) ord , and let Φ log : U log → (U log ) F be a Frobenius lifting. Let us consider the quotient Q (respectively, P) of E Z/p n Z by p n−1 · F 1 (E) (respectively, p n−1 · E). Thus, P is a quotient of Q, and P = E Z/p n−1 Z . Let T ⊆ P be the subbundle given us by the induction hypothesis. Let T ⊆ Q be the the inverse image of T ⊆ P via the surjection Q → P. 1 ∗ F 2 Then Φ ∗ (T ) F ⊆ Φ ∗ Q F deﬁnes a subbundle T Z/p n Z of F (E) Z/p n Z . It follows from the 1 1 2 deﬁnition of T F 2 p and the fact that we are on the ordinary locus that T Z/p n Z is ﬂat over 1 2 Z/p n Z. The existence of local horizontal generating sections for T Z/p n Z follows by taking such a section of T , lifting it to T , and then pulling back this lifted section of T to a 1 ∗ 2 section of T Z/p n Z via Φ . That the connection vanishes on this section follows from the deﬁnitions, plus the fact that pulling back by Φ adds an extra factor of p. Since T is Φ E - 1 1 n−1 2 2 Z = T . Thus, by the construction of T Z/p invariant, it follows that T Z/p n Z ⊗ Z/p n Z , 1 2 it is clear that T Z/p by Φ E will n Z is Φ E -invariant, since pulling back any lifting of T 1 2 give T Z/p n Z . Also, this same observation (coupled with the induction hypothesis) proves uniqueness. This completes the proof of the induction step. The last statement about 1 1 (T 2 ) ⊗2 follows from considering the splitting of the Hodge ﬁltration that T 2 deﬁnes. 126 Now suppose that our basepoint x : S → X maps into the ordinary locus X ord . Let Π ord = π 1 ((X log ) ord K , x K ). Thus, we have a natural morphism: Π ord → Π Let us denote the restriction of ρ to Π ord via this natural morphism by ρ ord : Π ord → GL ± (V ord ) Then if we apply the theory of [Falt], §2 to interpret Theorem 1.4, we see that the subbundle 1 T 2 ⊆ E| X ord in fact deﬁnes a sub-MF ∇ -object corresponding to an étale representation (ρ ord ) et : Π ord → GL ± (V et ) of Π ord , for some rank one free Z p -quotient module V ord → V et . Here, by “étale,” we mean that the kernel of (ρ ord ) et deﬁnes an étale covering of X ord . In other words, we have an exact sequence of (“up to {±1}”) representations of Π ord : 0 → V et ∨ (1) → V ord → V et → 0 where the “1” in parentheses is a Tate twist. We state this as a Corollary: Corollary 1.5. The restriction ρ ord of ρ to Π ord deﬁnes an (“up to {±1}”) module V ord of Π ord , which ﬁts into an exact sequence: 0 → V et ∨ (1) → V ord → V et → 0 where V et is étale and of rank one over Z p . The Canonical Frobenius Lifting over the Ordinary Locus In this subsection, we construct the generalized analogue (for an arbitrary canonical X ) of the p-adic endomorphism of the ordinary locus of the moduli stack of elliptic curves obtained by sending an elliptic curve with ordinary reduction to its quotient modulo its unique subgroup scheme which is étale locally isomorphic to μ p . In many respects, the construction is similar to (although not literally a logical consequence of) the construction ord of the Frobenius lifting on N g,r constructed in Chapter III, §2. log Consider the canonical indigenous bundle (E, ∇ E ) (of Theorem 1.1) on the canonical curve X log → S log . By Chapter II, Proposition 2.5, (E, ∇ E ) F p corresponds to an FL-bundle 0 → T F p → F → O X F p → 0 127 on X F log . By the material directly preceding Chapter II, Proposition 1.2, splittings of this p log exact sequence correspond to Frobenius liftings on X Z/p 2 Z . Now, over the ordinary locus log of X , the Hodge ﬁltration deﬁnes such a splitting. Let us denote the resulting Frobenius lifting on the ordinary locus by log ord F Φ 2 : (X log ) ord ) ) Z/p 2 Z Z/p 2 Z → ((X Let us denote by E the vector bundle which is the inductive limit of the following diagram: F 1 (E) ⏐ ⏐ p· −→ E F 1 (E) where the horizontal arrow is the natural inclusion. Note that E Z/p n Z depends only on F E Z/p n Z . By the deﬁnition of the renormalized Frobenius pull-back, Φ ∗ 2 E Z/p 2 Z is naturally isomorphic to E Z/p 2 Z . We shall identify these two sheaves in the following discussion. On the other hand, by considering the object in the upper right-hand corner of the diagram we obtain a morphism deﬁning E, F Φ ∗ 2 E Z/p 2 Z → E Z/p 2 Z ∗ 1 F ∗ 1 F whose restriction to Φ ∗ 2 F 1 (E) F Z/p 2 Z vanishes on p · Φ 2 F (E) Z/p 2 Z and maps Φ 2 F (E) Z/p 2 Z into p · F 1 (E) Z/p 2 Z (by the deﬁnition of the correspondence between Frobenius liftings and splittings of the FL-bundle F). log ord F Now let Ψ 3 : (X log ) ord ) ) Z/p 3 Z be any lifting of Φ 2 . Then, again from Z/p 3 Z → ((X the deﬁnition of the renormalized Frobenius pull-back, we obtain a morphism F Ψ ∗ 3 E Z/p 3 Z → E Z/p 3 Z which vanishes on p 2 · Ψ ∗ 3 F 1 (E) F Z/p 3 Z . However, if Ψ 3 is an arbitrary lifting of Φ 2 , then we ∗ 1 F don’t know that Ψ 3 F (E) Z/p 3 Z is mapped into F 1 (E) Z/p 3 Z . Now suppose that we modify Ψ 3 by a section δ ∈ Γ(X ord , T F p ). Let H T : T F p → (τ X log /S log ) F p | X ord be the isomorphism deﬁned by projecting to the Hodge ﬁltration. Then the subsheaf of E Z/p 3 Z given by the image of F 1 (E) F Z/p 3 Z under Ψ 3 + δ diﬀers from the corresponding image subsheaf under Ψ 3 by the amount H T (δ) ∈ Γ(X ord , (τ X log /S log ) F p ). Indeed, this follows from the deﬁnitions, plus the fact that the Kodaira-Spencer morphism for E is the identity. Since H T is an isomorphism, it thus follows that there exists a unique Frobenius lifting 128 log ord F Φ 3 : (X log ) ord ) ) Z/p 3 Z Z/p 3 Z → ((X 1 that lifts Φ 2 such that Φ ∗ 3 maps F 1 (E) F Z/p 3 Z into F (E). Clearly, we may repeat this procedure modulo p n for arbitrary n ≥ 3, so as to obtain a unique log ord ) → ((X log ) ord ) F Φ log X : (X such that under the natural morphism Φ ∗ X E F → E the Hodge ﬁltration is preserved. Note, moreover, that it follows from the fact that the Kodaira-Spencer morphism at the Hodge section is an isomorphism plus the interpretation of the FL-bundle F in terms of Frobenius liftings that this Frobenius lifting Φ log X is ordinary in the sense of Chapter III, Deﬁnition 1.1. In summary, we have proven the following result: Theorem 1.6. Let X log be a canonical curve; (E, ∇ E ) the canonical indigenous bundle on X log . Then there exists a unique ordinary Frobenius lifting (called canonical) log ord Φ log ) → ((X log ) ord ) F X : (X over the ordinary locus that preserves the Hodge ﬁltration. In particular, we can apply the theory of Chapter III, §1, to the Frobenius lift- ∇ ing Φ log X . Note that it follows immediately from the deﬁnitions that the MF -object (E, F 1 (E), ∇ E , Φ E )| (X log ) ord is precisely the uniformizing MF ∇ -object associated to Φ log X (as in Chapter III, Deﬁnition 1.3). Let us write T = (T 2 ) ⊗2 1 Thus, ∇ E (respectively, Φ E ) induces a natural connection (respectively, Frobenius action) on T , which deﬁnes the canonical tangential local system of Chapter III, Deﬁnition 1.2. Since T is a line bundle, it is the same to give (over an étale covering of X ord ) a generating Frobenius invariant section of it, or a generating Frobenius invariant section of its dual. Thus, (just as in Chapter III, Deﬁnition 1.11) if θ is such a section of T , then θ deﬁnes, at every z ∈ X ord (A) that avoids the marked points, a unique multiplicative parameter q θ ∈ R z × 129 (where R z is the completion of X ord at z). If the residue of θ is equal to one at a marked point z ∈ X ord (A), then we get a multiplicative parameter q ∈ R z (with valuation one at the divisor Im(z)) which is unique up to multiplication by a Te- ichmüller representative [k × ]. Deﬁnition 1.7. We shall call q (respectively, q θ ) the Serre-Tate parameter (respectively, relative to θ) at z. Note, in particular, that by the theory of Chapter III, §1, Φ −1 X maps q (respectively, q θ ) to p p q (respectively, q θ ). Remark. In some sense, it would be more aesthetically pleasing if one could obtain the Frobenius lifting of Theorem 1.6 in the following way. We consider the universal curve ord C → N g,r . Then C parametrizes (r + 1)-pointed stable curves of genus g, so we have a Frobenius lifting on some stack which is étale over C. If we could prove that this Frobenius ord lifting is compatible with the canonical Frobenius lifting on N g,r , then we could obtain a canonical Frobenius lifting on C (or at least some stack étale over C) simply by using the ord canonical Frobenius on N g,r+1 . The problem with this approach is that despite the fact that the canonical modular Frobenius liftings of Chapter III do have many interesting functorial relations (i.e., rela- tive to restriction to the boundary and log admissible coverings), in general, the sort of compatibility of Frobenius liftings necessary to make the above sketch of a proof work – namely, compatibility with “forgetting a marked point” – simply does not hold. Indeed, one can already see this in the case of the morphism M 1,1 → M 1,0 which is the identity on the underlying stacks, but which we think of as assigning to a one- pointed curve of genus one the underlying elliptic curve. Here, the canonical Frobenius on ord ord N 1,1 cannot be compatible with the canonical Frobenius on N 1,0 for the following reason. ord Since N 1,0 → M 1,0 is an open immersion, it would follow that the canonical Frobenius ord on N 1,1 would descend to an open formal subscheme of M 1,1 . But this would mean that even if a one-pointed curve of genus one in characteristic p belongs to several distinct quasiconformal equivalence classes (a phenomenon which by Chapter II, Proposition 3.13, does occur), the canonical liftings of that curve would be the same for all quasiconformal equivalence classes. But this would mean that we have several diﬀerent ordinary indigenous bundles on a single hyperbolic curve, all of which are Frobenius invariant. By Chapter III, Lemma 2.6, this is absurd. 130 §2. The Canonical Log p-divisible Group Although the existence of the canonical Galois representation of §1 is, in and of itself, of some interest, one technical drawback that it has is that it is diﬃcult to relate the log that it properties of the Galois representation or the characteristic zero coverings of X Q p log determines to X Z/p n Z . Thus, in this Section, we shall construct a log p-divisible group log on X which gives us back the canonical Galois representation (by looking at the Galois action on torsion points), but which has the advantage that one can study and understand its reductions modulo p n in a similar fashion to the elliptic modular case (which is studied in [KM]). Log p-divisible Groups at Inﬁnity We maintain the notation of the previous Section (although k need not be algebraically closed, just perfect). For basic facts about log schemes, we refer to [Kato] and [Kato2]. In [Kato2], certain ﬁnite, log ﬂat group objects over the compactiﬁed moduli stack of elliptic curves are introduced which are supposed to be the analogue at inﬁnity of the usual ﬁnite, ﬂat group schemes that one gets from elliptic curves by considering the kernel of multiplication by a power of p. Since we will use such objects (as well as the p-divisible group objects obtained by taking direct limits thereof) later in this Section, we take the time out in the present subsection to review explicitly the construction of these ﬁnite, log ﬂat group objects. Let R = A[[q]] be a complete local ring which is formally smooth of dimension one over A. If one inverts q, then by taking the (p n ) th root of q, one obtains an extension of ﬁnite ﬂat group schemes 0 → Z/p n Z(1) → G n → Z/p n Z → 0 over R[1/q]. Because q is not a unit in R, it is impossible to extend this extension of ﬁnite ﬂat group schemes over R[1/q] to an extension of ﬁnite ﬂat group schemes over R. Our goal in this subsection, however, is to exhibit a natural extension of the above exact sequence to an exact sequence deﬁned over R by working with group objects in the category of ﬁnite, log ﬂat log schemes over Spec(R) log . (In this subsection, we will regard Spec(R) as endowed with the log structure arising from the divisor deﬁned by q.) For nonnegative integers a, b, let M a,b be the monoid given by taking the quotient of N (where N is the monoid of nonnegative integers) by the equivalence relation generated by (p a , 0) ∼ (0, b). Let e 1 ∈ M a,b (respectively, e 2 ∈ M a,b ) be the image of (1, 0) (respec- tively, (0, 1)) in M a,b . Then it follows from the theory of [Kato2] (especially, §4.1, 5.1) that we can construct the desired extension 2 log 0 → Z/p n Z(1) → G n → Z/p n Z → 0 131 n as follows: For j ∈ {0, . . . , p n − 1}, consider the scheme G n,j given by R[x]/(x p − q j ), with the log structure given by the chart ([Kato], §2) M n,j with e 1 → x; e 2 → q. Denote log the resulting log scheme by (G ) log n,j . Let G n,j be the universal valuative log space ([Kato2], log §1.3.1) (which, in this case, will still be a log scheme) associated to (G ) log n,j . Let G n be log log the union of the G log n,j . Note that when we invert q, G n becomes G n . Endow G n with log the unique structure of group object that extends the group structure on G n . Then G n is a group object in the category of ﬁnite, log ﬂat log schemes over Spec(R) log , and it ﬁts into an exact sequence as above. log log As we allow n to vary, we get morphisms G n → G n+1 . Thus, we obtain an ind-group object G over Spec(R) log . Deﬁnition 2.1. We shall refer to G as the log p-divisible group over Spec(R) log obtained by taking p th power roots of q ∈ R. Finally, we remark that, although what we are doing here is, in some sense, just “trivial general nonsense,” its utility lies in the fact that by using it, we can obtain p-adic ﬁnite coverings of X log that are deﬁned over all of X log , thus allowing us to algebrize. Construction of the Canonical Log p-divisible Group We now turn to the construction of the canonical log p-divisible group on X log . Con- sider the MF ∇ -object (E, ∇ E , Φ E ), deﬁned by the canonical indigenous bundle. Let n ≥ 1. Let U ⊆ X be the open p-adic subscheme deﬁned by removing the marked points. Then the reduction modulo p n of (E, ∇ E , Φ E )| U deﬁnes, by [Falt], Theorem 7.1, a ﬁnite, ﬂat group scheme (annihilated by p n ), which we denote by G n | U → U . On the other hand, let R be the complete local ring at any one of the marked points. Then (E, ∇ E , Φ E ) deﬁnes a Serre-Tate parameter (as in Deﬁnition 1.7) q ∈ R/[k × ]. Let q ∈ R be any representative of q. Then R = A[[ q ]]. Let G log n | R be the log scheme constructed in the previous sub- n th section by taking a (p ) -root of q. Observe that diﬀerent choices of q give us naturally isomorphic G log , then G log n | R ’s. Also, note that if we invert q n | R becomes (G n | U )| R . Thus, log we see that G n | U and the various G n | R at the marked points glue together naturally to log form a ﬁnite, log ﬂat group object G log , which a priori is just p-adic, but may be n → X algebrized since X is proper over A. Also, as n varies, we obtain natural morphisms log . . . → G log n → G n+1 → . . . which thus form an inductive system of group objects. Deﬁnition 2.2. We shall call this inductive system of group objects the canonical log p-divisible group on X log . 132 Remark. As usual, strictly speaking we really have only deﬁned a “group up to {±1}.” That is, we really only have a group object over (perhaps) a ﬁnite étale covering of X of degree 4, plus descent data (satisfying the cocycle condition up to {±1}) down to the original X. We could, of course, develop the general nonsense of such “groups up to {±1},” but we choose not to, since it seems to serve no real purpose. If we invert p, then this log p-divisible group G log on X log deﬁnes a local system on log ) in free Z p -modules of rank two. Thus, we get a Galois representation the étale site (X Q p et on the Tate module T of characteristic zero torsion points of G log : ρ G log : Π → GL ± (T ) Then we have the following result (which is immediate from the theory of [Falt], especially the construction in the proof of Theorem 7.1): Proposition 2.3. The representation ρ G log is isomorphic to the canonical Galois repre- sentation ρ of Theorem 1.1. Review of the Theory of [Katz-Mazur] In this subsection, we apply to the log p-divisible group G log the theory of [KM], which is exposed in [KM] solely in the case of the canonical log p-divisible on the compact- iﬁed moduli stack of elliptic curves, but whose proofs go through without change for the canonical log p-divisible group G log on any canonical curve X log . First of all, because G log is a logarithmic p-divisible group, it follows from [Mess], Chapter II, Theorem 3.3.13, that if we consider the formal neighborhood of the identity section : X → G, we obtain a formally smooth formal scheme G( ) over X, which is easily seen to have relative dimension 1 over X. We would like to use this observation to apply the theory of [KM], Chapter 1, on “A-generators” and “A-structures” to G log . The theory there goes through just as in the modular case since the only technical assumption needed on the ﬁnite, ﬂat (logarithmic) group schemes whose A-generators we wish to parametrize is that they be closed subschemes of some smooth one-dimensional scheme. However, looking at the proofs of [KM], one sees that in fact, it suﬃces to have the ﬁnite, ﬂat (log) group schemes be closed subschemes of a formally smooth formal scheme (such as G( )) of relative dimension one. Thus, we can deﬁne various moduli problems, just as in [KM], Chapter 3, by means of various structures: (1) a Γ(n)-structure, which consists of giving a Drinfeld basis for G log n ; (2) a Γ 1 (n)-structure, which consists of giving a point “of exact order p n ” in G log n ; 133 (3) a Γ 0 (n)-structure, which consists of giving an isogeny G log → H log (where H log is also a log p-divisible group) whose kernel is cyclic of order p n . Moreover, just as in [KM], one proves that these various moduli problems are representable by schemes X(n); X 1 (n); X 0 (n) that are ﬁnite over X. Finally, all of these schemes X(n); X 1 (n); and X 0 (n) are, in fact, regular. Indeed, away from the marked points, the proofs of regularity in [KM], Chapters 5 and 6, boil down to general nonsense plus two technical results (Proposition 5.3.4 and Theorem 6.1.1). Since these technical results are proven, respectively, for arbitrary formal groups and arbitrary ﬁnite group schemes, it is immediate that the regularity proofs of [KM] in the modular case go through without change for X(n); X 1 (n); and X 0 (n). At the marked points, the combinatorial descriptions of the situation at the cusps in [KM], Chapter 10, go through without change for the above moduli problems. We thus obtain the following Theorem: Theorem 2.4. The schemes X(n); X 1 (n); and X 0 (n) that represent the moduli problems listed above are all regular, and hence equal to the normalizations of X in the ﬁnite cover- ings of X K deﬁned by the appropriate composites of ρ : Π → GL ± (V ) with ﬁnite quotients GL ± (V ) → G, just as in the classical modular case. Fix a positive integer n. We shall also need to analyze X 0 (n) modulo p, in a fashion similar to what is done in the modular case in [KM], Chapter 13. Let us (for the rest of the Section) denote X 0 (n) by Y , and let us use a subscript m on X, Y , etc., to denote reduction modulo p m+1 . Let us denote by Φ A : A → A, Φ k : k → k the respective absolute Frobenius morphisms, and by a superscript F m the result of base-changing an object by the m th power of Frobenius, and by Φ X 0 : X 0 → X 0 F the relative Frobenius of X 0 . Essentially, the description of Y 0 = Y ⊗ Z p F p given in [KM], Chapter 13, goes through in our situation here, but we need to do things with a little bit more care, since [KM] often falls back on the “crutch” of using the modular interpretation of their “X,” which we lack in this more general situation. For each ordered pair of nonnegative integers (a, b) such that a + b = n, we would like to deﬁne a k-scheme X 0 (a, b) of “(a, b)-cyclic isogenies” together with a k-morphism ι (a,b);0 : X 0 (a, b) → Y 0 . We do this as follows. If a, b ≥ 1, then we let X 0 (a, b) be the schematic inverse image of Inf p−1 (Δ) (the (p − 1) st inﬁnitesimal neighborhood of the diagonal) via b−1 F Φ a−1 X 0 × Φ X 0 : X 0 × X 0 a−b → X 0 F a−1 × X 0 F a−1 If a or b is zero, then we let X 0 (a, b) be the schematic inverse image of the diagonal Δ via Φ aX 0 × Φ bX 0 : X 0 × X 0 F 134 a−b a → X 0 F × X 0 F a Observe that in either case, (X 0 (a, b)) red is smooth over k; Φ aX 0 × Φ bX 0 maps X 0 (a, b) ⊆ a−b a a into Δ ⊆ X 0 F × X 0 F ; and X 0 (a, b) comes equipped with a ﬁnite, ﬂat, radicial X 0 × X 0 F morphism D 0 (a, b) : X 0 (a, b) → X 0 To deﬁne ι (a,b);0 : X 0 (a, b) → Y 0 we must specify a cyclic subgroup of order p n of D 0 (a, b) ∗ G log 0 . Now on the one hand, by composing the a th power of Frobenius with the b th -power of the Verschiebung (as in [KM], Theorem 13.3.5), we get some subgroup object of order p n of D 0 (a, b) ∗ G log 0 , and by the same argument as that given in [KM], Theorem 13.3.5, one sees that this subgroup must be cyclic (in the Drinfeldian sense). Thus, by the modular deﬁnition of Y 0 , we get a morphism ι (a,b);0 : X 0 (a, b) → Y 0 . In order to apply the theory of [KM], Chapter 13, we must verify the conditions (1) through (8) listed at the beginning of that Chapter. (Caution: The letters X and Y in [KM], Chapter 13, are used in the reverse way to the way that they are used here.) Conditions (1), (2), (4), (5), and (6) are trivial. Condition (3) follows from the regularity of Y and the fact that over a supersingular point, there is only one A-generator valued in k for a cyclic group, namely the identity element. Note that at ordinary points, one can do the same analysis of p th power isogenies of log p-divisible groups as is done in [KM], Chapter 13, §3. Thus, Condition (7) (that ι (a,b);0 is a closed immersion) and Condition (8) (that the ι (a,b);0 ’s deﬁne an isomorphism of the disjoint union of X 0 (a, b)’s with Y 0 over the ordinary locus) follow at the level of topological spaces from this analysis, and at the level of complete local rings by considering the deformation parameters for the domain and range log p-divisible groups of the isogeny. We thus get a result analogous to [KM], Theorem 13.4.7: Theorem 2.5. The k-scheme Y 0 is the disjoint union, with crossings at the supersingular points (in the terminology of [KM], Chapter 13, §1), of the n + 1 schemes X 0 (a, b) (where a + b = n). Let f (a,b) ∈ k[[x, y]] be the equation a−1 (x p a b−1 − y p ) p−1 b if a, b ≥ 1, and let it be x p − y p if a or b is zero. Then the completed local ring at a k-rational supersingular point of Y 0 is isomorphic to k[[x, y]]/( (a,b) 135 f (a,b) ) with the closed subscheme X 0 (a, b) ⊆ Y 0 given by the equation f (a,b) . §3. The Compactiﬁed Canonical Frobenius Lifting In this Section, we study the canonical Frobenius lifting on the ordinary locus of a canonical curve (deﬁned in Theorem 1.6). In particular, we study its behavior at su- persingular points, and “compactify it” in some sense, so as to obtain “pseudo-Hecke correspondences.” It is by abstracting the main properties of this compactiﬁed Frobenius in the canonical case that we shall obtain a geometric criterion for a curve to be canonical in §4. The Canonical Frobenius Lifting and the Canonical Log p-divisible Group Let us denote by log ord Φ log ) → ((X log ) ord ) F X : (X the canonical Frobenius lifting of Theorem 1.6. Let G log be the canonical log p-divisible group on X log of Deﬁnition 2.2. Then we rephrase Theorem 1.6 in terms of G log as follows: Theorem 3.1. The canonical Frobenius lifting of Theorem 1.6 log ord Φ log ) → ((X log ) ord ) F X : (X induces an isogeny of degree p Φ ∗ X (G log ) F | (X log ) ord → G log | (X log ) ord between the canonical log p-divisible groups that lifts the Frobenius morphism modulo p. log ord ) that has this property. Moreover, Φ log X is the unique Frobenius lifting over (X Proof. The existence of the isogeny follows from the fact that we have deﬁned a morphism between the respective Dieudonné crystals that respects the Hodge ﬁltrations. This induces the isogeny (see [BBM] and [Mess]). On the other hand, the uniqueness statement follows from the uniqueness statement in Theorem 1.6, together with the fact that if a Frobenius lifting induces such an isogeny, it automatically preserves the Hodge ﬁltrations on the Dieudonné crystals. 136 Let n ≥ 0. Let Y = X 0 (n). Let Y ord ⊆ Y be the p-adic open formal subscheme consisting of points lying over X ord . Now the Frobenius lifting of Theorem 3.1 allows us to extend the decomposition in characteristic p of Y F p into components corresponding to (a, b)-cyclic isogenies to a decomposition over A = W (k), on the ordinary locus. To obtain this decomposition, we deﬁne closed p-adic subschemes X(a, b) ord ⊆ X ord × A (X ord ) F a−b via the same recipe as we did for X 0 (a, b), except using our canonical Frobenius lifting log ord Φ log ) → ((X log ) ord ) F instead of Φ X 0 . Then, just as before, we get a natural X : (X embedding ι (a,b) : X(a, b) ord → Y ord (analogous to ι (a,b);0 ) which induces an isomorphism Y ord ∼ = X(a, b) ord (disjoint union) a+b=n Finally, over Y ord , we have a tautological isogeny G log → H Y log ord Y ord (where G log is the pull-back of G log to Y ord ) such that over X(a, b) ord , H Y log ord is naturally Y ord a−b a−b isomorphic to the pull-back of (G log ) F via the projection X(a, b) ord → (X ord ) F to the second factor. Local Analysis at Supersingular Points We now exploit the existence of the isogeny of Theorem 3.1 to understand the be- havior of the canonical Frobenius lifting at the supersingular points. Let x ∈ X(k) be a supersingular point. In studying x, we will often need to involve its various Frobenius i conjugates x F ∈ X(k) (which may be inﬁnite in number if the perfect ﬁeld k is not ﬁnite). We begin our analysis by considering the double iterate of the Frobenius morphism over some inﬁnitesimal neighborhood V ⊆ X F p at x: Φ 2 V : G log | V → G log | V F 2 Thus, V is the spectrum of a local artinian ring, with residue ﬁeld k. Let us assume that V is contained in the supersingular divisor (Chapter II, Proposition 2.6) of the canonical indigenous bundle. By deﬁnition, this means that over V , the Hodge ﬁltration coincides with the FL-bundle ﬁltration. It thus follows that over V , the kernels of the Verschiebung and Frobenius morphisms coincide. Since the kernel of the composite of the Verschiebung and the Frobenius is just the kernel of multiplication by p, it follows that the morphism Φ 2 V is isomorphic to the morphism “multiplication on p.” In particular, it follows that 137 G log | V F 2 ∼ = G log | V By iterating this isomorphism, we obtain that G log | V is isomorphic to the pull-back to V of a p-divisible group over k. Since the Kodaira-Spencer morphism of the Hodge ﬁltration of (E, ∇ E ) is an isomorphism, this implies that V must be Spec(k). Thus, the assumption that V lies inside the supersingular divisor implies that V is reduced. Put another way, we see that we have proven (in this general context) the analogue of Igusa’s theorem ([KM], p. 355): Proposition 3.2. The supersingular divisor of the canonical indigenous bundle (E, ∇ E ) is étale over k. Next, let us observe that for any x ∈ X(k), the completed local ring R x of X at x (which is formally smooth of dimension one over A) is naturally isomorphic to the universal deformation space of the p-divisible group G log | x . Indeed, it follows from the theory of [Mess] that deformations of G log | x are given by deformations of the Hodge ﬁltration; thus, our observation follows from the fact that the Kodaira-Spencer morphism of the Hodge ﬁltration of (E, ∇ E ) is an isomorphism. Now suppose that x ∈ X(k) is supersingular. Then the isomorphism G log | x F 2 ∼ = G log | x obtained above from the double iterate of Frobenius induces a natural isomorphism of complete local rings Ψ x : R x F 2 ∼ = R x which will play an important role in the sequel. Now ﬁx a number n ≥ 1, and let Y = X 0 (n). If x ∈ X(k), let us denote by X x the formal spectrum of X at x, i.e., Spf(R x ). We will use similar notation for Y . Over Y , we have a tautological cyclic isogeny of order p n : log G log Y → H Y Fix a supersingular x ∈ X(k). By the analysis of [KM], reviewed in §2, there exists a unique y ∈ Y (k) lying over x. Now by thinking of the completed local rings of X as universal classifying spaces, we obtain a morphism: (D y , R y ) : Y y → X x × A X x F n 138 where D y is the classifying morphism for the “domain p-divisible group” G log Y , and R y is log the classifying morphism for the “range p-divisible group” H Y . Here we use the fact that restricted to x, the tautological isogeny is just the n th iterate of the Frobenius morphism, so H Y log | y = G log . Also, note that by the deformation theory of [Mess], a deformation of the x F n tautological isogeny is uniquely determined by the induced deformations of the domain and range p-divisible groups. It thus follows that the morphism (D y , R y ) is formally unramiﬁed, hence a closed immersion. Thus, henceforth, we shall think of Y y as a formal divisor inside X x × X x F n by means of the closed immersion (D y , R y ). One of the most important properties of this divisor Y y is its symmetry. More precisely, given Y y , one can obtain a divisor in X x F n × X x in two ways: (1) by applying the isomorphism X x × X x F n ∼ = X x F n × X x given by switch- ing the two factors; (2) by conjugating ﬁrst by the n th power of Frobenius, so as to obtain a divisor in X x F n × X x F 2n and then applying the isomorphism Spf(Ψ x ) −n to the second factor. Then we claim that these two divisors in X x F n × X x are the same. Indeed, to see this, it suﬃces to trace what happens to the tautological isogeny. Let us consider the second procedure stated above. First, we conjugate the tautological isogeny by Frobenius: n F → (H Y log ) F (G log Y y ) y n Since at y, this isogeny is just the n th iterate of the Frobenius morphism, by looking at Dieudonné modules, it follows that the kernel of this isogeny is contained in the kernel of multiplication by p n . Thus, we get a morphism n F (H Y log ) F → (G log Y y ) y n n 2n F Since (H y log ) F = (G log , we thus see that the divisor in X x F n × X x obtained this y ) way is just the divisor of isogenies (lifting the n th iterate of Frobenius) from the universal deformation of G log x (pulled back from the second factor) to the universal deformation of log F n (G x ) (pulled back from the ﬁrst factor). On the other hand, it we look at the divisor obtained from the ﬁrst procedure stated above, it admits exactly the same description. This proves the claim. The next important property of this divisor Y y ⊆ X x × X x F n is that if we restrict it to the ordinary locus, it becomes equal to the union of the “local versions” of the schemes X(a, b) ord (described at the end of the preceding subsection). More precisely, X(a, b) ord a−b is deﬁned as a closed subscheme of X ord × (X ord ) F . Thus, we obtain X(a, b) ord | X x ⊆ a−b (X ord × (X ord ) F ) X x by restricting to the formal scheme X x . Since a − b ≡ a + b = 139 n(mod 2), by applying the appropriate power of Ψ x , we thus obtain a subscheme, which we shall call n ord X(a, b) ord × (X ord ) F ) X x x ⊆ (X Then it follows from the functorial deﬁnition of Y (in terms of parametrizing isogenies) that Y y | X ord = X(a, b) ord x (disjoint union) a+b=n Let us denote the supersingular divisor of X by X ss . We are now ready to summarize what we have done: Deﬁnition 3.3. We shall call the pair def H x = (Y y ; Ψ x ) consisting of the divisor Y y ⊆ X x × X x F n and the isomorphism Ψ x the n th canonical local Hecke correspondence of X log at x. We shall call the triple (Y → X; Φ log X ; {H x } x∈X ss ) the n th canonical pseudo-Hecke correspondence of X log . Of course ideally, the local Hecke correspondences H x would glue together to form a n global Hecke correspondence, i.e., a morphism Y → X × X F , just as in the classical case with modular curves. We shall investigate this issue in the next subsection, but (unfortu- nately) what we shall ﬁnd is that the existence of such global Hecke correspondences is a rather rare phenomenon. Global Hecke Correspondences We maintain the notation of the previous subsection. Deﬁnition 3.4. We shall say that the canonical curve X log admits a global n th canonical Hecke correspondence if there exists a morphism (D, R) : Y → X × X F 140 n that is equal to the local morphisms (D y , R y ) of the preceding subsection for every y ∈ Y lying over a supersingular point of X. We say that X log is of Hecke type if it admits a global n th canonical Hecke correspondence for every n ≥ 1. 2 Proposition 3.5. Suppose that there exists an isomorphism Ψ log : X log → (X log ) F that induces the morphism Spf(Ψ x ) when localized at every supersingular x ∈ X. Then X log is of Hecke type. Proof. This follows immediately from gluing together Y ord with the various Y y ’s by means of Ψ log . Corollary 3.6. Suppose that X log , along with all of its supersingular points are deﬁned over F p 2 . Then X log is of Hecke type. Proof. Over F p 2 , one can take Ψ log to be the identity, for (by functoriality) the Ψ x ’s must all be the identity. The classical example of a case where X log is of Hecke type is the case where X log = log M 1,0 , the compactiﬁed moduli stack of elliptic curves (over Z p ). This case is studied in detail in [Shi], Chapter 3. To see that the supersingular points are deﬁned over F p 2 , one 2 repeats the argument preceding Proposition 3.2, to obtain an isomorphism E F ∼ = E for every supersingular elliptic curve E. Suppose that X log is canonical of Hecke type. Then we remark that just as in the log ⊗N classical case, one can deﬁne Hecke operators on H 0 (X, (ω X/S ) ) Q p (as well as one the log étale cohomology of X Q , etc.). Moreover, (by the same proofs as in the classical case) p the eﬀect on q-expansions (where q is the Serre-Tate parameter at a marked point) is the same as in the classical case. (See, e.g., [Lang] for more details.) Next, we justify the assertion (made in the preceding subsection) that, in some sense, being of Hecke type is a rather rare phenomenon. Indeed, if X log admits a global ﬁrst canonical Hecke correspondence, consider its reduction modulo p: (D, R) F p : Y F p → X F p × X F F p Let Z = Y F p . Then Z = Z Z , with Z mapping isomorphically to X F p via D F p , and −1 D F p | Z isomorphic to the Frobenius morphism from Z ∼ = X F F p to X F p . On the other −1 hand, R F p must map Z isomorphically to X F F p . Thus, Z is isomorphic to both X F F p 2 and X F F p . In particular, X F p ∼ = X F F p , i.e., the moduli of X F p are deﬁned over F p 2 , which is a very rare phenomenon. 141 Remark. At the present time, the author does not know of any canonical X log of Hecke type, except for those that arise directly from the modular case. §4. p-adic Green’s Functions In this Section, we give a geometric criterion for a curve to be canonical: namely, the existence of a Frobenius lifting of the right height, over an open p-adic formal subscheme of the curve, with “nice behavior” at the points where it is not deﬁned. We will make these terms precise below, but the point of interest is that this criterion does not depend ord on knowing the action of the canonical Frobenius Φ N on N g,r – that is, it is intrinsic to the curve – and, moreover, it is not phrased in terms of indigenous bundles. Now in the case of elliptic curves (regarded parabolically), the canonical lifting deﬁned in terms of indigenous bundles is the same as the canonical lifting deﬁned in Serre-Tate theory (one of the deﬁnitions of which is the existence of a global Frobenius lifting). Thus, it is interesting to note that the existence of an “admissible Frobenius lifting” amounts to just the existence of a Frobenius lifting (in the case of elliptic curves). In other words, one may regard the geometric criterion given here as the proper hyperbolic generalization of the statement that an elliptic curve (whose reduction modulo p is ordinary) is Serre-Tate canonical if and only if it admits a Frobenius lifting. Compactiﬁed Frobenius Liftings In this subsection, motivated by the construction of the pseudo-Hecke correspondences in the previous Section, we deﬁne the general notion of a “compactiﬁed Frobenius lifting.” Let X log → S log be a smooth r-pointed curve of genus g. Let φ log : Y log → X log be a ﬁnite, ﬂat morphism such that Y is regular (necessarily of dimension two), and the log structure on Y log is deﬁned by a divisor on Y which is étale over A and equal to the set-theoretic inverse image of the divisor of marked points of X. Let U ⊆ X be an open formal subscheme that contains all the marked points. We endow U with the log structure pulled back from X log , and call the resulting log formal scheme U log . We shall use the notation X x , Y y , etc. (as in the previous subsection) to denote the formal schemes which are the formal neighborhoods of the closed points x, y, etc. Suppose that Φ log : U log → (U log ) F is a Frobenius lifting. Deﬁnition 4.1. We shall say that (φ log : Y log → X log ; ι) is a naive compactiﬁcation of the Frobenius lifting Φ log if, when we take V = φ −1 (U ), the following conditions are satisﬁed: 142 (1) V log splits as a disjoint union (V log ) (V log ) ; (2) φ log | V : (V log ) → U log is an isomorphism; (3) ι log : (V log ) → (U log ) F −1 (4) φ log | V ◦ (ι log ) −1 : (U log ) F is an isomorphism; −1 → U log is the morphism (Φ log ) F We shall frequently identify V and U , and V and U F −1 −1 . . Suppose that (φ log ; ι log ) is a naive compactiﬁcation for Φ log . Note that φ is necessarily of degree p + 1. Also, note that V admits a canonical embedding ι (1,0) : V → U × U F where we take ι (1,0) = (id, Φ), while V admits an embedding ι (0,1) : V → U × U F where we take ι (0,1) = ((Φ) F −1 −1 , id). Let x ∈ X(k). Let Ψ x : R x F 2 ∼ = R x be an isomorphism. Suppose that y ∈ Y (k) maps to x. Let (D y , R y ) : Y y → X x × X x F be a closed embedding, where D y is obtained by restricting φ. def Deﬁnition 4.2. We shall say that H x = (Ψ x ; (D y , R y )) is a local compactiﬁcation for the Frobenius lifting Φ log at x (relative to (φ log ; ι log )) if (1) y is the unique closed point of Y lying over x; (2) the divisor Y y → X x × X x F is symmetric in the sense that the two divisors that it induces in X x F × X x (by switching and by Frobenius- conjugating, then applying Ψ x ) are the same; (3) the restriction of (D y , R y ) to V is the union of ι (1,0) , and (ι (0,1) ) F composed with Spf(Ψ x ) −1 × (id). 143 2 Note that by the ﬁrst condition, x cannot lie in U . Now let us consider (Y y ) Z/p 2 Z . Since Y y ⊆ X x × X x F , it follows that (Y y ) Z/p 2 Z is Spf of a local ring of the form: def R = (A/p 2 A)[[ξ, η]]/ψ(ξ, η) where ξ is a local parameter for (X x ) Z/p 2 Z , and η = ξ F is the Frobenius-conjugate local parameter for (X x F ) Z/p 2 Z . Since Y y → X x is ﬂat, the last condition implies that (Y y ) F p has exactly two irreducible components, both of which are reduced. Thus, if we denote by a “bar” the reduction of functions modulo p, we see that ψ is a product of two distinct prime elements of k[[ξ, η]]. In fact, we can say more. Outside the special point of (Y y ) F p , p these two primes deﬁne the closed subschemes ξ − η and η p − Ψ x (ξ f = ξ p − η; F 2 ). Let 2 g = η p − Ψ x (ξ F ) Thus, we may assume that ψ = f · g. In other words, we can write ψ = f · g + π where π ∈ p · k[[ξ, η]]. In fact, π is actually p times a unit in k[[ξ, η]], since Y y is regular. So far we have been working with functions on (Y y ) Z/p 2 Z . Now let us restrict to functions on the open formal subscheme D(g) ⊆ (Y y ) Z/p 2 Z (i.e., where g is invertible). Thus, we are in eﬀect restricting to the graph of Φ. Let us denote the restriction morphism on functions by ζ : R → R[1/g]. Then we obtain, in R[1/g], ζ(η) = ζ(ξ) p + ζ(π) · ζ(g) −1 By interpreting this open formal subscheme D(g) as the graph of Φ, this tells us that Φ −1 (ξ F ) is a function which is not regular at x, but has a pole of order one (since g has a zero of order one). In particular, it tells us that the Frobenius lifting Φ does not admit a regular extension to any neighborhood of x. We summarize this as follows: Proposition 4.3. If H x is a local compactiﬁcation of Φ log at x, then (Y y ) F p is a node, and x ∈ / U . Also, the Frobenius lifting Φ does not admit an extension to any neighborhood of x. Deﬁnition 4.4. We shall call C = (φ log ; ι log ; {H x } x ∈U / ) 144 a compactiﬁcation of the Frobenius lifting Φ log if (φ; ι) is a naive compactiﬁcation of Φ log , and for each x ∈ / U , we are given a local compactiﬁcation H x of Φ log relative to (φ; ι) (where k in the deﬁnition above is replaced by the ﬁeld of rationality of x). Thus, in particular, by what we did in the last two subsections, Proposition 4.5. Suppose that X log is a canonical curve. Then its ﬁrst canonical pseudo-Hecke correspondence is a compactiﬁcation of the canonical Frobenius lifting on (X log ) ord . def log Suppose that (φ log ; ι log ; {H x } x ∈U . Let us consider Z = / ) is a compactiﬁcation of Φ Y F p . It follows from the above deﬁnition that Z is reduced and has exactly two irreducible components Z and Z with V F p ⊆ Z ; V F p ⊆ Z . Since Z is geometrically connected, smooth, proper, and birationally equivalent to X F p over k, it follows that Z ∼ = X F p . ∼ F −1 Similarly, Z = X F p . Moreover, except at the points of intersection of Z and Z (which are nodes), Z is smooth over k. Proposition 4.6. (Assuming that X log is hyperbolic) Y must be connected. Proof. It suﬃces to prove that Z is connected. Suppose that Z is not connected. Then φ Z : Z → X F p is ﬁnite and birational, hence an isomorphism. It thus follows that Z lifts to a connected component Y of Y such that φ Y : Y → X is an isomorphism. −1 On the other hand, Z is proper and smooth over k, and birational to X F F p , hence −1 Z ∼ = X F , and φ| Z : Z → X F is the Frobenius morphism. Moreover, Z lifts to a F p p connected component Y of Y . Thus, φ| Y : Y → X is a Frobenius lifting. But if X log is hyperbolic, such Frobenius liftings cannot exist, for the nonzero morphism of line bundles log → ω Y log /S violates degree restrictions. (φ| Y ) ∗ ω X/S The Height of a Frobenius Lifting Finally, we note that often it is useful to have a precise measure of how far a Frobenius lifting fails to extend over all of X. For this, we introduce the notion of the height of a Frobenius lifting, as follows. Let F → X F p be the T F p -torsor of Frobenius liftings on open sub-log schemes of Thus, if Φ log : U log → (U log ) F is a Frobenius lifting, its reduction modulo p 2 deﬁnes a section σ Φ : U F p → F of this torsor. Let P be the projective bundle that canonically compactiﬁes F. Thus, P → X F p is a P 1 -bundle. Recall the notion of the canonical height of a section of P → X F p , introduced at the beginning of Chapter I, §2. Since X F p is proper over k, it follows that σ Φ extends uniquely to a section σ Φ : X F p → P . We now make the following log X Z/p 2 Z . 145 Deﬁnition 4.7. We deﬁne the height ht(Φ) of the Frobenius lifting Φ log to be the canonical height of the section σ Φ of P → X F p . More concretely, the height of Φ log can be deﬁned as follows. If x ∈ X F log , then let t be a p log local parameter of X at x. Let Ξ x be a local Frobenius lifting deﬁned in a neighborhood −1 of x. Then p 1 (Ξ −1 (t)) is a rational function δ x on X F log . Let us say that the local x (t) − Φ p log height ht x (Φ) of Φ at x is: (1) equal to 0 if this function δ x is regular at x; (2) equal to the order of the pole of δ x at x otherwise. Then we have the formula: Proposition 4.8. We have p [k(x) : k] ht x (Φ) ht(Φ) + (2g − 2 + r) = 2 log x∈X F p Proof. This follows immediately from considering the intersection number of σ Φ with the “section at inﬁnity” given by the complement of F in P . Corollary 4.9. If Φ log admits a compactiﬁcation, then the local heights at points outside U are all one. Thus, p ht(Φ) = − (2g − 2 + r) + deg k (X − U ) F p 2 where we regard (X − U ) F p as having the reduced, induced scheme structure. Proof. The statement about local heights follows from the explicit computation preceding Proposition 4.3. The following is the main result of this subsection: Proposition 4.10. If Φ log has height ≤ 1−g− 12 r, then P (with its connection ∇ P induced by that of F) is a nilpotent, admissible indigenous bundle. In particular, if (P, ∇ P ) is also ordinary, then X log is isomorphic to a canonical curve modulo p 2 , and Φ log is equal to the canonical Frobenius lifting (of Theorem 3.1) modulo p 2 . 146 Proof. Indeed, suppose that ht(Φ) ≤ 1 − g − 12 r. Consider the Kodaira-Spencer morphism of the section σ Φ of P → X. By the general properties of FL-bundles (Chapter II, §1), we know that the Kodaira-Spencer morphism cannot vanish (for the section at inﬁnity of P → X is the unique horizontal section). But by degree considerations (i.e., the assumption on ht(Φ)), once the Kodaira-Spencer morphism is nonzero, it must be an isomorphism. It thus follows that P (with its connection induced by that of F) is a nilpotent, admissible indigenous bundle. The last statement follows from the construction of the canonical lifting and the canonical Frobenius. Thus, we see that the compactiﬁed Frobenius liftings that we are really interested in are the ones that “look nice modulo p:” Deﬁnition 4.11. A compactiﬁed Frobenius lifting C = (φ log ; ι log ; {H x } x ∈U / ) is called admissible if (1) ht(Φ) = g − 1 + 12 r; (2) the associated (P, ∇ P ) (as in Proposition 4.10) is ordinary; (3) the reductions modulo p of the isomorphisms Ψ x (that make up H x ) are equal to the canonical “Ψ x ” of Deﬁnition 3.3. Note that for an admissible compactiﬁed Frobenius C, all the objects involved (that is, Φ log ; φ log : Y log → X log ; ι log ; Ψ x ; (D y , R y )) are completely determined modulo p (up to isomorphism) once one ﬁxes the supersingular divisor (X − U ) F p . Or, in other words, Proposition 4.12. An admissible compactiﬁed Frobenius C on X log determines a p-adic quasiconformal equivalence class α to which X log belongs. If two admissible compactiﬁed Frobenii C and C on X log determine the same α, then, modulo p, all the objects that make up C are isomorphic to those that make up C . Admissible Frobenius Liftings Since an admissible compactiﬁed Frobenius is determined modulo p by the p-adic quasiconformal equivalence class α, the next step is to understand what the possible de- formations looks like. Let C be an admissible compactiﬁed Frobenius, and let us consider C Z/p 2 Z , i.e., the reductions modulo p 2 of all the objects involved. Suppose we start with 147 the data (Φ log ) Z/p 2 Z ; (Ψ x ) Z/p 2 Z (for all supersingular x). Now it is easy to see (by looking at points valued in an arbitrary scheme) that the symmetry condition on the divisor Y y amounts to the statement that (Ψ x ) Z/p 2 Z restricted to the ordinary locus commutes with Φ Z/p 2 Z . Let β be an automorphism of (X x ) Z/p 2 Z which is equal to the identity modulo p. Then we claim that if Φ Z/p 2 Z commutes with β, then β is the identity. Indeed, since derivations on X F p act trivially on functions that are p th powers, we get that Φ Z/p 2 Z ◦ β = Φ Z/p 2 Z Thus, β F ◦ Φ Z/p 2 Z = Φ Z/p 2 Z which implies that β is the identity, since Φ Z/p 2 Z is faithfully ﬂat. Thus, in summary, the (Ψ x ) Z/p 2 Z are determined uniquely by the condition that they commute with Φ Z/p 2 Z . Next, we consider Y log . We can break Y log up into three parts: (V log ) ; (V log ) ; and the Y y ’s. Since (V log ) and (V log ) are determined up to natural isomorphism by X log , it remains to determine the Y y ’s, and the gluing morphisms. But Y y and its gluing morphisms to (V log ) and (V log ) are completely speciﬁed once one knows the divisor Y y in X x ×X X F . Moreover, it follows from the condition (3) of Deﬁnition 4.2, that this divisor is determined by Φ and Ψ x . But we just saw that (Ψ x ) Z/p 2 Z is determined by Φ Z/p 2 Z . Thus, we conclude 2 that (H x ) Z/p 2 Z and φ log Z/p 2 Z are completely determined by Φ Z/p Z . Sorting through all the deﬁnitions, we thus see that we have proven that C Z/p 2 Z is entirely determined by the p-adic quasiconformal equivalence class α and the deformation Φ Z/p 2 Z . Moreover, there is nothing special about working modulo p 2 : the same arguments can be made modulo an arbitrary power of p. Thus, we see that we have proven the following result: Lemma 4.13. Let C be an admissible compactiﬁed Frobenius lifting on X log . Then C Z/p n Z is completely determined by Φ log Z/p n Z . This Lemma suggests the following deﬁnition: Deﬁnition 4.14. Let α be a p-adic quasiconformal equivalence class to which X log belongs. Let U ⊆ X be the ordinary locus for α. Let Φ log : U log → (U log ) F be a Frobenius lifting over U log . Then we shall say that Φ log is an admissible Frobenius lifting for (X log , α) if it arises from a (necessarily unique) admissible compactiﬁed Frobenius C. Thus, we can can regard admissible Frobenius liftings as being Frobenius liftings (over the ordinary locus) that happen to have special behavior near the supersingular points. 148 Next let us consider two admissible Frobenius liftings Φ log and (Φ log ) on the same curve X log . Let us suppose that they are equal modulo p n (where n ≥ 2). In this discussion, we shall always be working modulo p n+1 , so (by abuse of notation) we shall use Φ log and (Φ log ) to denote the respective reductions modulo p n+1 . Now just as in the discussion preceding Proposition 4.3, for Φ log , (Y y ) Z/p n+1 Z is Spf of a local ring of the form: def R = (A/p n+1 A)[[ξ, η]]/ψ(ξ, η) where ξ is a local parameter for (X x ) Z/p n+1 Z , and η = ξ F is the Frobenius-conjugate local parameter for (X x F ) Z/p n+1 Z . Moreover, we may assume that ψ = f · g + π where f = ξ p − η; 2 g = η p − Ψ x (ξ F ) and π ∈ p · (A/p n A)[[ξ, η]]. In fact, π is actually p times a unit, since Y y is regular. Similarly, for (Φ log ) , we have R = (A/p n+1 A)[[ξ, η]]/ψ (ξ, η) def with ψ = f · g + π , and g = η p − Ψ x (ξ F ). Thus, 2 g ≡ g (mod p n ); π ≡ π (mod p n ) So far we have been working with functions on (Y y ) Z/p n+1 Z . Now let us restrict to the open subscheme corresponding to the graph of Φ (or Φ ). Let us denote the restriction morphism on functions by ζ. Then we obtain, on this open subscheme, in the unprimed case: ζ(η) = ζ(ξ) p + ζ(π) · ζ(g) −1 and, in the primed case: ζ(η) = ζ(ξ) p + ζ(π ) · ζ(g ) −1 = ζ(ξ) p + ζ(π ) · ζ(g) −1 since the diﬀerence between g and g becomes zero when multiplied by p. Ultimately, we are interested in computing the diﬀerence between the two Frobenius liftings Φ and Φ . 149 That is, we wish to understand the diﬀerence between where η is taken by the two liftings. But by the above formulas, the diﬀerence is of the form: {ζ(π) − ζ(π )}(ζ(g)) −1 Moreover, because the diﬀerence in brackets is divisible by p n , only the residue of g modulo p is involved in the above expression. Since this residue has a zero of order exactly one at x, it follows that the diﬀerence between the two Frobenius liftings – which forms a section of T F p over X ord – has poles of order at most one at the supersingular points. Since the morphism H T F p : T F p → (τ X log /S log ) F p (given by composing the p-curvature of (P, ∇ P ) with the projection given by the Hodge ﬁltration) has zeroes at the supersingular points, it thus follows that the diﬀerence between Φ log and (Φ log ) deﬁnes a global section of (τ X log /S log ) F p . Since we are dealing with hy- perbolic curves, though, (τ X log /S log ) F p has negative degree, hence has no global sections. Thus, Φ log = (Φ log ) . In summary, we have proven the following strengthened form of Lemma 4.13: Lemma 4.15. If there exists an admissible Frobenius lifting of (X log , α), then it is unique, and contains no nontrivial deformations modulo any power of p. Geometric Criterion for Canonicality Let us ﬁx X log → S log , a smooth r-pointed curve of genus g, and a p-adic quasicon- formal equivalence class α to which X log belongs. Let Φ log be an admissible Frobenius lifting for (X log , α). log → S log is also a smooth r-pointed curve of genus g such that Now suppose that X log ( X ) F p = X F p . The following “Rigidity Lemma” is fundamental to this subsection: log log , α) admits an admissible Frobenius lifting (Φ log ) . Then Lemma 4.16. Suppose that ( X log = X log . X Proof. We propose to prove inductively (on n) the following statement: log , (Φ log ) ) coincides with (X log , Φ log ) modulo p n+1 . (*) ( X We know that this statement holds for n = 0. Assume that it holds for n − 1. Consider log log log the diﬀerence between the deformations X Z/p n+1 Z and X Z/p n+1 Z of X Z/p n Z . It deﬁnes a class 150 μ ∈ H 1 (X F p , (τ X log /S log ) F p ) Let C (respectively, C ) be the compactiﬁed Frobenius corresponding to Φ log (respectively, log log ) Z/p n Z , we can consider the diﬀerence between the defor- (Φ log ) ). Since Y Z/p n Z = (Y log log log mations Y Z/p ) Z/p n+1 Z of Y Z/p n Z . This gives us a class n+1 Z and (Y ν ∈ Ext 1 Y F p (Ω Y log /k , O Y F p ) F p Since φ Z/p n Z = φ Z/p n Z , the pull-back map deﬁned by either of these morphisms gives us log a map φ −1 : φ ∗ (ω X/S ) F p → Ω Y log /k , which is generically zero over the component of Y F p F p that we called Z in the proof of Proposition 4.6. Now φ −1 induces a pull-back morphism on global Ext’s ψ : Ext 1 Y F p (Ω Y log /k , O Y F p ) → H 1 (Y F p , φ ∗ (τ X log /S log ) F p ) F p The condition that the morphism φ Z/p n+1 Z deform compatibly with the deformation of log , (Y ) log ) Z/p n+1 Z to a morphism φ n+1 is exactly that (X log , Y log ) Z/p n+1 Z to ( X Z Z/p ψ(ν) = μ| Y F p Note that this condition, as well as the cohomology modules in which μ and ν live, log log are independent of n. Thus, by adding μ and ν to φ log Z/p 2 Z : Y Z/p 2 Z → X Z/p 2 Z , we obtain a new morphism φ log : Y log → X log (of Z/p 2 Z-ﬂat schemes). Then, by restricting φ log to the open subscheme of Y log deﬁned by the ordinary locus of Z , we obtain a Frobenius lifting Ξ log on the ordinary locus of X log . The only points at which Ξ is not deﬁned are the supersingular points (determined by α). Moreover, by the calculation of the discussion preceding Proposition 4.3, it follows that the local height of Ξ log at a supersingular point is ≤ 1. Indeed, in the notation of loc. cit., “π” is equal to p times an element of k[[ξ, η]], which, this time, might not be a unit since Y log might not be the reduction modulo p 2 of a regular scheme; hence the inequality ≤ 1, rather than the sharp equality = 1. At any rate, it thus follows that ht(Ξ) ≤ ht(Φ Z/p 2 Z ). But then, by Proposition 4.10, X log is equal to some canonical curve reduced modulo p 2 , and, by Proposition 2.6, (4), of Chapter II, it log thus follows that this canonical curve is the one determined by α. Thus, X log = X Z/p 2 Z . log log n+1 and X coincide modulo p . By Lemma 4.15 But this means that μ = 0, so X log (the rigidity of an admissible Frobenius lifting), it thus follows that Φ and (Φ log ) also coincide modulo p n+1 . This proves the induction step, and hence the Lemma. Putting everything together, we see that we have proven the following geometric criterion for a curve to be canonical: 151 Theorem 4.17. Let X log → S log be a smooth r-pointed curve of genus g. Let α be a p-adic quasiconformal equivalence to which X log belongs. Then X log is canonical if and only if (X log , α) admits an admissible Frobenius lifting. Proof. We saw in §3 that a canonical curve admits an admissible Frobenius lifting. On the other hand, given an (X log , α) which admits an admissible Frobenius lifting, there exists log ≡ X log modulo p) which is canonical, hence admits an admissible log , α) (with X an ( X log . Frobenius lifting. Thus, by Lemma 4.16, it follows that X log = X Deﬁnition 4.18. Suppose that (X log , α) admits an admissible Frobenius lifting Φ log . Then we shall call Φ log the p-adic Green’s function. Remark. The justiﬁcation for this terminology is as follows. In the classical complex case, one of the main approaches to proving that hyperbolic curves can be uniformized by the upper half plane is given by constructing a Green’s function on the universal covering space of the Riemann surface (see, e.g., [FK]). Once one proves that the universal covering space is just the upper half plane, then one sees that this Green’s function is really just the logarithm of the hyperbolic distance between two points. On the other hand, the canonical Frobenius lifting Φ log may also be regarded as giving us a notion of distance on X log . Indeed, in the classical modular case, where X log parametrizes elliptic curves, if one can get from point a to point b by applying Φ log a total of N times, then it means that the corresponding elliptic curves are isogenous via a cyclic isogeny of order p N . Thus, the analogy between Φ log and the classical complex Green’s function (which is just the logartihm of the hyperbolic distance) will be established once one accepts that isogeny is the proper analogue of distance. But to see this, one need merely think of lattices in Q 2 p , which one can draw schematically as a graph. Then two lattices are related by an isogeny of order p N if and only if they are N edges apart on this graph. This establishes the relationship between isogeny and distance. Remark. We also observe that for elliptic curves (regarded parabolically), the same deﬁ- nition of compactiﬁed Frobenius liftings, and admissible Frobenius liftings goes through, but everything is trivial, since there are no supersingular points to contend with. Thus, we (trivially) obtain the analogue of Theorem 4.17: that an elliptic curve over A with ordinary reduction is canonical if and only if it admits a Frobenius lifting deﬁned everywhere on the curve. 152 Chapter V: Uniformizations of Ordinary Curves §0. Introduction Having studied the case of canonical curves in the previous Chapter, in this Chapter we turn to the case of arbitrary curves with ordinary reduction modulo p. We do this by working with the universal case: i.e., the universal curve over the moduli stack. Unlike the canonical case, one does not quite obtain such objects as the canonical Galois rep- resentation or the canonical log p-divisible group over the given base. Instead, one must pass to various “schemes of multiplicative periods” – i.e., certain inﬁnite coverings of the original base – in order to obtain such objects. On the other hand, since these objects are canonically associated to the curve over the given base, it is natural to guess that they should descend from the scheme of multiplicative periods back down to the original base in some appropriate sense. The key idea here is that, for instance in the case of the canonical Galois representation (which is fundamental to the construction of all the other objects), if one works with modules of rank two, not over Z p , but over some appropriate ring of p-adic periods D Gal , then one can in fact construct a canonical Galois representation over the original base. Thus, one obtains a representation of the entire arithmetic fundamental Gal group into GL ± ), which in some sense extends the representation of the geometric 2 (D fundamental group into GL ± 2 (Z p ). Moreover, (in the hyperbolic case) this representation Gal ) is canonical, and dual crystalline in of the arithmetic fundamental group into GL ± 2 (D some appropriate sense, despite the fact that (unlike the case handled in [Falt], §2), it is on a space of inﬁnite rank over Z p . The process of passing from the canonical representa- tion of the geometric fundamental group into GL ± 2 (Z p ) to the canonical representation of Gal the arithmetic fundamental group into GL ± (D ) is a sort of crystalline analogue of the 2 notion of an induced representation in group theory. We therefore refer to this process as the process of crystalline induction. Once one has this canonical dual crystalline representation of the arithmetic funda- Gal mental group into GL ± ), one can linearize the obstruction to extending the repre- 2 (D sentation of the geometric fundamental group into GL ± 2 (Z p ) to the full arithmetic funda- mental group. This linearization tells one, for instance, that as soon as one can extend the representation of the geometric fundamental group into GL ± 2 (Z p ) at all to the arith- metic fundamental group, this extension is automatically dual crystalline. This procedure of linearizing the obstruction also allows one to see that this obstruction is precisely the hyperbolic analogue of the obstruction to splitting a certain exact sequence of p-adic local systems on the moduli stack of ordinary elliptic curves (in the parabolic case). §1. Crystalline Induction In Chapter III, we constructed a Frobenius-invariant indigenous bundle on the univer- ord sal curve over N g,r . Unfortunately, unlike the case of a canonical curve, such information 153 does not immediately constitute an object of the category MF ∇ (see [Falt], §2), so we cannot immediately convert it into a Galois representation. The problem is that our con- nection on the indigenous bundle is only a relative connection (for the universal curve over ord N g,r ), not a full connection on the total space of the universal curve. Also, the obstruction to extending it to a full connection on the total space of the universal curve is nonzero. Thus, in order to obtain a Galois representation, we must replace the indigenous bundle by a certain natural “thickening” of the indigenous bundle. This thickening formally car- ries the structure of an object of the category MF ∇ , but has the disadvantage of being of inﬁnite rank, so that we cannot immediately apply the theory of [Falt] to this object. Fortunately, it is not diﬃcult to extend the theory of [Falt] so as to handle such objects of inﬁnite rank. We thus obtain a Galois representation, as desired, which turns out to be a sort of crystalline analogue of the notion of an “induced representation” in group theory. The Crystalline-Induced MF ∇ -object def Let p be an odd prime. Let S be formally smooth over A = W (k), where k is a perfect ﬁeld of characteristic p. Let us assume that S is endowed with a log structure induced by a relative divisor with normal crossings over W (k). Let S log be the resulting log formal scheme. Let f log : X log → S log be an r-pointed stable curve of genus g. Also, let us assume that the classifying morphism S → M g,r deﬁned by f log is étale. Let Φ log : S log → S log be an ordinary Frobenius lifting. Now let us suppose that our indigenous bundle (E, ∇ E ) on X log is invariant under the renormalized Frobenius (Chapter III, Deﬁnition 1.4): that is, Φ E : (E, ∇ E ) ∼ = F ∗ S log (E, ∇ E ) Φ where the superscript “Φ” denotes pull-back by Φ, and the subscript “S log ” denotes that we are considering the relative renormalized Frobenius pull-back over S log . (We shall denote by F ∗ A the renormalized Frobenius pull-back over A.) Let D S be the quasi-coherent O S -algebra (with the O S -action from the right) which is obtained by taking the p-adic completion of the PD-envelope of the diagonal embedding of S log in S log × A S log (see, e.g., [Kato], §5.8). Thus, D S has an ideal I S ⊆ D S with D S /I S = log log O S , and I S /I S 2 ∼ = Ω S/A = Ω S . Note that D S has a natural logarithmic connection ∇ D S . We shall regard D S as a ﬁltered object with connection, whose ﬁltration is given by [i] F i (D S ) = I S (i.e., divided powers of I S ). Thus, the Kodaira-Spencer morphism for the log subquotient of the ﬁltration given by F 1 (D S )/F 2 (D S ) = I S /I S 2 ∼ = Ω S is the identity map. Note that the Frobenius lifting Φ log on S log induces a morphism Φ D S : Φ ∗ D S → D S which preserves the Hodge ﬁltration. Finally, let us denote by ((D S ) X , ∇ (D S ) X ) the pull-back of (D S , ∇ D S ) to X log . Next, let us consider the obstruction to deﬁning a full logarthmic connection ∇ on E (i.e., relative to X log → Spec(A)) with the following properties: 154 (1) ∇ has trivial determinant; (2) the restriction of ∇ to a relative connection (for X log → S log ) is ∇ E ; (i.e., the (3) the curvature of ∇ is an Ad(E)-valued section of ∧ 2 Ω log S log log ω X/S ⊗ O S Ω S -part of the curvature vanishes). It is easy to see that the obstruction class to deﬁning such a connection is a section η E of log 0 1 R 1 f DR,∗ Ad(E) ⊗ O S Ω log S = F (R f DR,∗ Ad(E) ⊗ O S Ω S ) log and Φ E whose projection to R 1 f ∗ τ X log /S log ⊗ O S Ω log S is the identity. Also, note that Φ log 1 induce a Frobenius action on R f DR,∗ Ad(E) ⊗ O S Ω S , which, by naturality, preserves η E . Thus, in particular, we see that unless we modify E in some way, there is no hope of constructing a full connection ∇ as speciﬁed above. PD Thus, we make the following construction. Let us write S log × S log for Spf(D S ) (where we take “Spf” with respect to the p-adic topology). Similarly, we shall write PD PD PD S log × S log × S log , X log × X log , etc. for the obvious p-adic completions of PD- PD envelopes at the respective diagonals. Also, we have two projections π R , π L : S log × S log → S log to the left and right by which we can pull-back X log → S log to obtain curves PD PD (X log ) L → S log × S log , and (X log ) R → S log × S log . Moreover, both of these curves PD form PD-thickenings of X log → S log → S log × S log (where the second morphism is the diagonal embedding). It thus follows that if we pull-back (E, ∇ E ) to obtain an indigenous PD bundle on the curve (X log ) L → S log × S log , this indigenous bundle deﬁnes a crystal PD on Crys(X log /(S log × S log )) which we can then evaluate on the thickening (X log ) R to obtain a rank two vector bundle E (on X R ). If we then push this sheaf E forward via the projection X R → X, we obtain a quasi-coherent sheaf E D on X. Moreover, E D has the structure of a D S -module, hence of a (D S ) X -module. In fact, E D is a locally free (D S ) X - module of rank two. Moreover, E D is equipped with a natural Hodge ﬁltration compatible with that of (D S ) X . Next, we would like to equip E D with a full logarithmic connection that is compatible with its structure as a (D S ) X -module and the connection on (D S ) X . First, note that PD PD PD PD X log × S log × S log → S log × S log × S log is a PD-thickening of X log → S log → PD PD S log × S log × S log (where the second morphism is the diagonal embedding). Thus, if we PD PD PD pull-back (E, ∇ E ) to X log × S log × S log , we obtain a crystal E on Crys(X log /(S log × PD PD PD PD PD S log × S log )). On the other hand, S log × X log × X log → S log × S log × S log is also PD PD a PD-thickening of X log → S log → S log × S log × S log . Thus, if we evaluate E on PD PD PD PD S log × X log × X log , and then push forward via the projection S log × X log × X log → 155 PD PD X log × X log , we obtain a sheaf E on X log × X log . Now since (D S ) X is equipped PD with a connection, the two pull-backs of (D S ) X to X log × X log via the two projections PD X π R , π L X : X log × X log → X log can be identiﬁed; we denote the resulting sheaf of algebras PD on X log × X log by (D S ) PD X log × X log . Then E is equipped with the structure of a locally free -module of rank two. On the other hand, from the deﬁnition of E D , it follows PD X log × X log X ∗ that both (π L X ) ∗ E D and (π R ) E D are naturally isomorphic (as (D S ) -modules) to PD X log × X log PD X ∗ E , hence to each other. This isomorphism (π L X ) ∗ E D ∼ ) E D on X log × X log deﬁnes = (π R a full logarithmic connection ∇ E D on E D (with respect to X log → Spec(A)). Moreover, (D S ) one checks easily that this connection is integral. Finally, we have a Frobenius action Φ E D : F ∗ S log (E D ⊗ D S ,Φ D S D S ) ∼ = E D Here, in the deﬁnition of F ∗ S log , we ﬁrst pull back the relevant crystal by means of relative Frobenius, and then consider the subsheaf consisting of sections whose reduction modulo p is contained in the subsheaf of Φ ∗ X F p (E D ⊗ D S ,Φ D S D S ) F p ∼ = (Φ ∗ X F p E F p ) ⊗ O S D S given by (Φ ∗ X F p F 1 (E) F p ) ⊗ O S D S ⊆ (Φ ∗ X F p E F p ) ⊗ O S D S Theorem 1.1. Over X log , there exists a natural, locally free, rank two (D S ) X -module E D equipped with a Hodge ﬁltration, a full integrable logarithmic connection ∇ E D (relative to X log → Spec(A)), and a Frobenius action Φ E D : F ∗ S log (E D ⊗ D S ,Φ D S D S ) ∼ = E D such that E D ⊗ D S (D S /I S ) = E. The Ring of Additive Periods Before we can convert the induced object of Theorem 1.1 into a Galois representation, we must ﬁrst study the Galois representation associated to D S , with its natural ﬁltration, connection, and Frobenius action. Once we have done this, since E D is of ﬁnite rank over 156 D S , converting E D into a Galois representation will be no more diﬃcult than the “classical case” discussed in [Falt], §2. Let us ﬁrst note that, just as when we constructed “canonical aﬃne coordinates” in Chapter III, §1, by considering the slopes of the Frobenius action Φ D S , we obtain a unique Frobenius-equivariant embedding of O S -modules Ω log S → I S whose composite with the projection I S → I S /I S 2 = Ω log S is the identity. It is here that we use the divided powers of I S ⊆ D S . Let us write Geo(D S ) for the subbundle of D S generated by Ω log S and O S . Note that Geo(D S ) is stabilized by ∇ D S and by Frobenius. Moreover, the Hodge ﬁltration on D S induces a Hodge ﬁltration on Geo(D S ). Observe that with this extra data, Geo(D S ) becomes isomorphic to the uniformizing MF ∇ -object associated to Φ log (of Deﬁnition 1.3 of Chapter III). Now we want to pass to Galois representations. Let us assume that we have chosen (once and for all) a base-point of S that avoids the divisors deﬁning the log structure. In the following, our fundamental groups will be with respect to this base-point. Since our construction will be canonical, we can work étale locally on S. Thus, we can assume that S is aﬃne. We may also assume that S log is small (in the sense of [Falt], §2): that is, S log is log étale over A[T 1 , . . . , T d ] (with the log structure given by the divisor T 1 · . . . · T d ). We shall call these parameters T 1 , . . . , T d small parameters. Then we would like to consider the ring B + (S log ) of [Falt], §2. We will not review the deﬁnition of this ring here, since it is rather involved, but roughly speaking, it is obtained by log (1) taking the normalization S of S log in the maximal covering of S K which is étale in characteristic zero; (2) reducing S modulo p and taking its perfection; (3) taking the Witt ring with coeﬃcients in this perfection; (4) adjoining the divided powers of a certain ideal to this Witt ring; and (5) ﬁnally, completing with respect to a certain topology. In particular, (1) B + (S log ) is obtained as the inverse limit of a projective system of PD- thickenings of the O S F p -algebra S F p ; (2) B + (S log ) has an ideal I + ⊆ B + (S log ) which is Galois-invariant and such that B + (S log )/I + ∼ = S ∧ (i.e., the p-adic completion of S). 157 Moreover, B + (S log ) comes equipped with a natural Frobenius action (which we shall denote log by means of a superscripted “F ”), as well as a continuous π 1 (S K )-action, which commutes + log with the Frobenius. The Frobenius invariants of B (S ) are given by Z p ⊆ B + (S log ). There is a Galois equivariant injection β : Z p (1) → B + (S log ). Frequently, we shall abuse notation and write β ∈ B + (S log ) for the element of B + (S log ) obtained by applying β to some generator of Z p (1). Then the Frobenius action on β takes β to p · β. We will denote by B(S log ) the ring obtained from B + (S log ) by inverting β and p. This completes our review of B + (−). Now let us return to the speciﬁc situation we have at hand. By thinking of (D S , ∇ D S ) as a crystal, and using the fact that B + (S log ) is an inverse limit of PD-thickenings of a certain O S F p -algebra, one can evaluate this crystal on B + (S log ) (and complete p-adically) to obtain a B + (S log )-module which we shall denote by D S ⊗ O S B + (S log ) (where the “hat” denotes p-adic completion). Alternatively, one can embed O S into B + (S log ) by means of a choice of small parameters, and then take the literal tensor prod- uct, as described in [Falt], §2. In our situation, however, since we are given a Frobenius lifting Φ log , the most useful point of view will be to embed O S → B + (S log ) O S into B + (S log ) by means of the Frobenius lifting Φ log . Indeed, the choice of Frobenius lifting gives us an embedding of O S into the ring of Witt vectors that appears in the construction (reviewed above) of B + (S log ). Then, we may regard the module considered above as obtained via the literal tensor product with respect to this particular embedding of O S into B + (S log ). At any rate, D S ⊗ O S B + (S log ) has a natural ﬁltration and Frobenius action. Let T log → S log be the ﬁnite covering deﬁned by Φ log . (Thus, T log ∼ = S log .) Then log D S ⊗ O S B + (S log ) also has a natural action by π 1 (T K ). The reason why we must restrict log ), is that the way the Galois action is deﬁned to T log , rather than considering all of π 1 (S K (see [Falt], §2) involves exponentiating the connection ∇ D S , so in order for the exponential series to converge, one must be in a situation where the connection acts in a suﬃciently log ) nilpotent fashion. For convenience, let us write Π T log (respectively, Π S log ) for π 1 (T K log (respectively, π 1 (S K )). Let us recall the uniformizing Galois representation P et (Deﬁnition 1.4 of Chapter III) associated to the ordinary Frobenius lifting Φ log . Recall that P et ﬁts into an exact sequence of Π T log -modules 0 → Θ et Φ (1) → P et → Z p → 0 The space of splittings of this sequence then forms an aﬃne Z p -scheme, which is the ∨ ∨ ). More concretely, Aﬀ(P et ) is a polynomial ring over Z p in spectrum of some ring Aﬀ(P et 158 3g − 3 + r variables which is equipped with an action by Π T log . Moreover, the submodule ∨ ∨ ∨ → Aﬀ(P et ). We shall refer to Aﬀ(P et ) as the of polynomials of degree ≤ 1 is given by P et ∨ aﬃnization of P et . ∨ ∨ Note that Aﬀ(P et ) has a natural Π T log -invariant augmentation Aﬀ(P et ) → F p . Let Gal ∨ D S be the p-adic completion of the PD-envelope of Aﬀ(P et ) at this augmentation. Thus, D S Gal is equipped with a natural structure of Π T log -algebra, and, moreover, we have a Π T log -invariant inclusion ∨ → D S Gal P et In other words, D S may be identiﬁed with the ring of additive periods (of Deﬁnition 1.5 of Chapter III). On the other hand, since P et is the Galois representation contravariantly associated to Geo(D S ), it follows that we have a morphism P et → Geo(D S ) ∨ ⊗ O S B + (S log ) which respects the Hodge ﬁltrations, Frobenius actions (where P η is endowed with the trivial Hodge ﬁltration and Frobenius action), and Galois actions (by Π T log ). “Switching duals,” we thus see that we have a morphism ∨ Geo(D S ) → P et ⊗ Z p B + (S log ) → D S Gal ⊗ Z p B + (S log ) which respects Hodge, Frobenius, and Galois. Next, since Geo(D S ) generates D S as a “PD-polynomial algebra” with no relations, it thus follows that we obtain a morphism D S → D S Gal ⊗ Z p B + (S log ) which respects Hodge, Frobenius, and Galois. Finally, tensoring with B + (S log ), we obtain the following result: Proposition 1.2. We have a morphism D S ⊗ O S B + (S log ) → D S Gal ⊗ Z p B + (S log ) which respects Hodge ﬁltrations, Frobenius actions, and Galois actions (by Π T log ). The Crystalline-Induced Galois Representation Let U log ⊆ X log be a small aﬃne open subset. Choose a Frobenius lifting Ψ log on U log that is compatible with Φ log on S log . Thus, Ψ log gives us an embedding of O U into B + (U log ) which ﬁts into a commutative diagram 159 O S ⏐ ⏐ −→ B + (S log ) ⏐ ⏐ O U −→ B + (U log ) We would like to consider ∨ ⊗ O U B + (U log ) G U = E D def where the “∨” denotes the dual as a D S -module. The problem is to show that G U has enough Frobenius-invariant sections in the zeroth step of its ﬁltration. The reason that we cannot apply the theory of [Falt], §2 directly is that the relevant theorem (Theorem 2.4 of loc. cit.) assumes a bound on the number of steps in the ﬁltration of the MF ∇ -object under consideration. On the other hand, E D has inﬁnitely many steps, What we can do is base-change G U by the morphism of Proposition 1.2. We then obtain a free (D S Gal ⊗ Z p B + (U log ))-module G U of rank two. Moreover, G U is equipped with a Galois action, a Hodge ﬁltration, and a Frobenius action F ∗ ((G U ) F ) ∼ = G U (where the superscripted “F ” denotes base-change by the Frobenius on B + (U log )). Proposition 1.3. The submodule (F 0 (G U )) F =1 (consisting of Frobenius-invariant elements of F 0 (G U )) forms a free D S Gal -module E U Gal of rank two. Proof. The proof is entirely the same as that of Theorem 2.4 of [Falt], §2. The point of base-changing by the morphism of Proposition 1.2 is that this enables us to replace ∨ whose Hodge ﬁltrations have inﬁnitely many steps by objects like G U objects like E D whose Hodge ﬁltration has essentially only two steps. In fact, over T log , the relative connection ∇ E on E actually extends to a full connection ∇ modulo p, so (E F p , ∇ ) deﬁnes a Galois representation onto some F p -vector space E (⊆ E F ∨ p ⊗ B + (U log ) F p ) of dimension two. Moreover, G U has a ﬁltration (deﬁned by taking divided powers of the augmentation 160 ideal of D S Gal → F p ) whose subquotients are tensor products of E F ∨ p ⊗ B + (U log ) F p with symmetric powers of Ω et (−1) F p . That is to say, we know that F 0 (−) F =1 for all of these subquotients is as desired, so next we want to consider the issue of whether the various extensions involved split over B + (U log ). But this issue is precisely that discussed in the proof of Theorem 2.4 of [Falt], §2. Thus, we see that we have enough Frobenius invariants, at least over the PD-completion of D S Gal . But it is a simple exercise to see that the fact that the original Frobenius action is deﬁned over D S Gal (i.e., not just over the PD-completion of D S Gal ) implies that the Frobenius invariants will also be deﬁned over D S Gal itself. This completes the proof. def Let X T log = X log × S log T log . Similarly, we have U T log ⊆ X T log . Now we have a natural π 1 ((U T log ) K )-action on G U Since this action preserves the ﬁltration and commutes with Frobenius, we thus get an action of π 1 ((U T log ) K ) on E U Gal , which is compatible with mul- tiplication by elements of D S Gal and the Π T log -action on D S Gal . Moreover, as we vary the open subset U ⊆ X, the resulting E U Gal ’s are clearly compatible. Thus, they glue together to form a π 1 ((X T log ) K )-D S Gal -module Gal E X We state this as a Theorem: Theorem 1.4. The crystalline-induced MF ∇ -object (E D ; F i (E D ); Φ E D ; ∇ E D ) of Theorem 1.1 has associated to it an (up to ±1) π 1 ((X T log ) K )-D S Gal -module Gal E X Gal which is a free D S Gal -module of rank two. We shall refer to E X as the crystalline-induced ∇ Galois representation associated to the induced MF -object of Theorem 1.1. Remark. Unlike the case of canonical curves, where one actually has a dual crystalline representation (in the sense of [Falt], §2) into GL ± 2 (Z p ), in the case of noncanonical curves, Gal E X is as close a p-adic analogue as one can get to the canonical representation in the complex case. In the following subsection, we shall make the phrase “as close as one can get” more precise. 161 Relation to the Canonical Aﬃne Coordinates log Let T ∞ → S log be the inverse limit of the coverings obtained by iterating Φ log . Let log log log be the pull-back to T ∞ of X log → S log . Let us choose base-points once and X T ∞ → T ∞ for all, and let def Π 1 = π 1 ((X T log ) K ); def Π ∞ = π 1 ((X T log ) ) ∞ K Thus, Π ∞ ⊆ Π 1 is a closed subgroup. In the preceding subsection, we constructed a D S Gal - Gal Π 1 -module which we called E X . If we restrict our Galois representations from Π 1 to Π ∞ , then we obtain a Π ∞ -equivariant surjection π S : D S Gal → Z p whose kernel is the augmentation ideal I S Gal , i.e., the PD-ideal generated by Ω Gal S . Thus, if Gal we base change E X by π S , we obtain a rank two Z p -module with a continuous Π ∞ -action which we denote by: def Gal ⊗ D Gal ,π S Z p E 0 = E X S Thus, in summary, E 0 has the advantage that it is of rank two over Z p , but the disadvantage Gal that it only has a Π ∞ - (as opposed to a full Π 1 -) action, while E X has the advantage that is has a natural Π 1 -action, but the disadvantage that it is of rank two over the rather large ring of additive periods D S Gal . In this subsection, we show that the canonical aﬃne coordinates (of Chapter III, Theorem 3.6) measure precisely the degree to which the Π ∞ action on E 0 cannot be extended to a full action of Π 1 . We begin with the following fundamental observation. Let Δ ⊆ Π ∞ ⊆ Π 1 be the log ). Let geometric fundamental group, i.e., the kernel of the natural morphism Π 1 → π 1 (T K us consider the exact sequence of Π ∞ -modules Gal ⊗ D Gal D S Gal /(I S Gal ) 2 → E 0 → 0 0 → E 0 ⊗ Z p I S Gal /(I S Gal ) 2 → E X S By considering the underlying Δ-module structures, we obtain that the above exact se- quence deﬁnes an extension class η Gal ∈ H 1 (Δ, Ad(E 0 )) ⊗ Z p I S Gal /(I S Gal ) 2 = H 1 (Δ, Ad(E 0 )) ⊗ Z p Ω Gal S log which is ﬁxed by the natural action of π 1 ((T ∞ ) K ) on this cohomology group. On the other hand, because our original indigenous bundle is ordinary, we see that we have a log ) K )-equivariant inclusion π 1 ((T ∞ 162 ∨ 1 (Ω Gal S ) → H (Δ, Ad(E 0 )) Then we claim that η Gal is precisely the class in H 1 (Δ, Ad(E 0 )) ⊗ Z p Ω Gal that corresponds S to this inclusion. Indeed, this follows immediately from observing that η Gal is essentially the Galois version of the class η E (the obstruction to the existence of a full connection on E) that we encountered on our way to constructing E D . It then follows immediately from the way one passes from MF ∇ -objects to Galois representations that η Gal is the above inclusion, as claimed. This observation concerning η Gal will be the fundamental “hard fact” underlying what we do in this subsection; the rest will be general nonsense. The general nonsense that we will use is the theory of [Schl]. Let us denote the Π ∞ -module E 0 ⊗ Z p F p by (E 0 ) F p . Note that since over T log , the obstruction to putting a full connection on E F p vanishes, so we get a genuine MF ∇ -object (whose underlying vector bundle is (E D ⊗ D S /I S ) F p ) modulo p. Thus, the Π ∞ -action on (E 0 ) F p extends to a natural (dual crystalline) action of Π 1 on (E 0 ) F p . We apply Schlessinger’s theory to the functor on artinian rings B with residue ﬁeld F p that assigns to such a ring B the set of isomorphism classes of continuous representations of Δ on a free B-module E B of rank two such that (E B ) ⊗ B F p = (E 0 ) F p . Since H i (Δ, Ad((E 0 ) F p )) is zero if i = 1, and of dimension 6g − 6 + 2r over F p if i = 1, it follows easily from [Schl] that this functor is prorepresented by a formal scheme R over Z p . Moreover, R is formally smooth over Z p , of relative dimension 6g − 6 + 2r. Now we claim that there is a natural continuous action of π 1 ((T log ) K ) on R. Indeed, let α ∈ Π 1 . Since Δ ⊆ Π 1 is a normal subgroup, for any representation of Δ on some E B as above, we obtain a new representation by conjugating elements of Δ by α, and then acting on E B in the original fashion. Since the original Δ-action on E B extends to a full Π 1 -action on (E B ) ⊗ B F p , it follows that this new representation is isomorphic to the old after base change by B → F p . Thus, the new representation deﬁnes a new B-valued point of R. This deﬁnes an action of Π 1 on R which is clearly trivial on Δ ⊆ Π 1 . Thus, we obtain a natural action of π 1 ((T log ) K ) on R. Gal Let us now turn to applying R to understanding the Π 1 -module E X . Let D S Gal be Gal the p-adic completion of D S Gal . The underlying Δ-module structure on E X deﬁnes a classifying morphism κ : Spf( D S Gal ) → R (This is O.K. despite the fact that D S Gal ⊗ Z p Z/p N Z is not artinian, since Δ is topologically ﬁnitely generated.) Let σ 0 : Spf(Z p ) → R be the composite of Spf(π S ) with κ. Let R PD be the p-adic completion of the PD-envelope of R at σ 0 . Let 163 κ PD : Spf( D S Gal ) → R PD be the morphism induced by κ. Then the morphism induced by κ PD on the Zariski tangent spaces at σ 0 is precisely the injection corresponding to the class η Gal considered above. Thus, we have the following: Lemma 1.5. The morphism κ PD is a closed immersion of formal schemes. Gal actually comes from Now let us consider the fact that the Δ-module structure on E X Gal a Π 1 -module structure which is D S -semilinear (with respect to the Π 1 -action on D S Gal log through π 1 (T K )). If we translate this statement by means of the functorial interpretation of R, we obtain the following: log log )-equivariant with respect to the natural π 1 (T K )- Lemma 1.6. The morphism κ is π 1 (T K Gal actions on D S and R. log Now let φ : Γ → π 1 (T K ) be a continuous homomorphism of topological groups. Let 1 → Δ → Π Γ → Γ → 1 be the pull-back of the group extension log 1 → Δ → Π 1 → π 1 (T K ) → 1 by means of φ. Then one can consider the issue of whether or not the Δ-action on E 0 extends to a continuous, Z p -linear action of Π Γ on E 0 . Note that since H 0 (Δ, Ad(E 0 )) = 0, as long as we require that the associated determinant representation of Π Γ is the cyclotomic character, such an extension will always be unique (up to ±1). On the other hand, by the same reasoning as that used in Lemma 1.6, the Δ-action on E 0 will extend to a Π Γ -action on E 0 if and only if the Z p -valued point σ 0 of R is ﬁxed by Γ (acting through φ). Moreover, by the preceding two lemmas, we see that σ 0 is ﬁxed by Γ if and only if the Z p -valued point of D S Gal deﬁned by π S is stabilized by Γ. But this, in turn, is equivalent to the statement that the restriction of the canonical extension class η Φ (discussed just before Deﬁnition 1.5 of Chapter III) becomes trivial when restricted to Γ. Now let us suppose that (B, m B ) is a local ring with residue ﬁeld k which is p-adically complete, Z p -ﬂat, and has a topologically nilpotent PD-structure on m B . (Note that for such a ring B, log : (1+m B ) → m B and exp : m B → (1+m B ) deﬁne inverse isomorphisms.) Let Γ = π 1 (Spf(B) K ). Suppose further that ψ : Spf(B) → S 164 is a morphism whose image avoids the divisor deﬁning the log structure on S log . Then for log ) which is some closed subgroup Γ ⊆ Γ of ﬁnite index, we have a morphism φ : Γ → π 1 (T K compatible with the morphism induced on fundamental groups by ψ. Recall from Chapter III, Theorem 3.8, the canonical aﬃne coordinates corresponding to ψ. These coordinates are valued in B (or, more precisely, in m B ). Moreover, they are zero if and only if the class η Φ becomes zero when restricted to Γ . It is easy to see that η Φ | Γ = 0 if and only if η Φ | Γ = 0. Also, we know from Chapter III that ψ corresponds to a canonical curve if and only if the canonical aﬃne coordinates are zero. Thus, putting everything together, we obtain the following result: Theorem 1.7. The morphism ψ corresponds to a canonical curve if and only if the Δ-action on E 0 extends to a Z p -linear, continuous action of Π Γ on E 0 . More generally, but less precisely, we see that: (*) The canonical aﬃne coordinates in m B associated to ψ are a mea- sure of the obstruction to extending the Δ-action on E 0 to a Z p -linear, continuous action of Π Γ on E 0 . Since the class η Φ is “as nonzero as it can be” on S log , we thus see that we have justiﬁed Gal the statement made at the end of the preceding subsection that the Π 1 -module E X is “as close as one can get” to extending the Δ-action on E 0 to a full Π 1 -action. Remark. In some sense, we can describe what we have done in this subsection as follows. Consider the obstruction to extending the Δ-action on E 0 to an action of Π Γ . A priori, this obstruction is highly nonlinear and diﬃcult to get one’s hands on explicitly. Note that this nonlinearity exists despite the fact that we already have a Π ∞ -action on E 0 , and the discrepancy between Π 1 and Π ∞ is “essentially” a linear Z p -space of rank 3g−3+r. Rather, the nonlinearity arises fundamentally from the fact that we are considering representations into GL ± 2 , which is not an abelian (or even solvable) group. In particular, the moduli space R of representations of Δ into GL ± 2 has no natural linear structure. Thus, the point of Gal Gal constructing E X and reinterpreting the existence of E X in terms of R, as we have done in this subsection, was to linearize this obstruction by means of the uniformization of (the relevant part of) R PD by means κ PD . Remark. For the reader interested in pursuing analogies with the complex case, we also make the following observation. Since R is the local moduli space of deformations of the canonical representation Δ → GL ± 2 (Z p ) of the geometric fundamental group, it is natural to regard R as a sort of local p-adic analogue of the space R C of isomorphism classes of representations of the geometric fundamental group into PSL 2 (C) in the complex case. Thus, R C has complex dimension 6g − 6 + 2r. Inside R C , one has Fricke space R R , with real dimension 6r − 6 + 2r, corresponding to the representations into PSL 2 (R). In a neighborhood of the canonical representation of a curve, R R maps diﬀeomorphically onto 165 M g,r . Thus, one can regard the subspace Spf( D S Gal ) → R PD as analogous to Fricke space R R ⊆ R C in the complex case. The Parabolic Case Before proceeding, we pause to take a brief look at what happens for elliptic curves (regarded parabolically). Although there are many similarities between the parabolic and hyperbolic cases, there are also certain diﬀerences. This is not so surprising if one considers the canonical representations arising from uniformizations in the complex case. Indeed, in the complex context, for hyperbolic curves, the canonical representation of the funda- mental group into PSL 2 (R) is completely well-deﬁned up to conjugation by an element of PSL 2 (R), while for elliptic curves, the morphism (induced by deck transformations on the uniformization by C) gives a representation of the fundamental group into the group of translations G a (C) of the complex plane, but this representation is not well-deﬁned up to conjugation by an element of G a (C). Rather, there is an ambiguity of multiplication by a complex number. On the other hand, this same phenomenon of “lack of rigidity” ultimately is a con- sequence of the overall linearity of the situation, which has positive aspects, as well. For instance, one can carry out the construction of the induced MF ∇ -object (Theorem 1.1) for elliptic curves just as in the hyperbolic case. However, precisely because the obstruction to deﬁning an MF ∇ -object is entirely linear from the outset, this approach is a sort of overkill. Thus, in the following we propose to examine the obstruction to deﬁning a “full” MF ∇ -object (i.e., the same obstruction as the one we examined in the hyperbolic case) directly, at the level of Galois representations, without resorting to the tool of crystalline induction. Thus, let us assume that f log : X log → S log is an ordinary zero pointed curve of genus one such that the classifying morphism S → M 1,0 is étale. Since the canonical representation should be an extension of a rank one étale representation by its dual Tate log log twisted once, we consider the local system R 1 (f K ) et,∗ Z p (1) on S K . Let us denote the def log )-module corresponding to this local system by H 1 . Then there exists a Π S = π 1 (S K rank one Π S -submodule E(1) ⊆ H 1 such that the action of Π S on E is unramiﬁed, and we have an exact sequence: 0 → E(1) → H 1 → E ∨ → 0 Suppose that E is an étale Π S -module such that (E ) ⊗2 = E ∨ . Then, the canonical representation of the universal elliptic curve should be an extension of E by (E ) ∨ (1). If we tensor the above exact sequence by E, we get an exact sequence of Π S -modules 0 → E ⊗2 (1) → H 1 ⊗ E → Z p → 0 Thus, the obstruction to the existence of such a canonical representation is precisely the obstruction to lifting 1 ∈ Z p in the above exact sequence. This obstruction class lies in 166 H 1 (Π S , E ⊗2 (1)) and coincides with the class η Φ deﬁned by the canonical Frobenius lifting Φ log on S log . That is to say, we end up with essentially the same conclusion as in the hyperbolic case: Namely, that the obstruction to the existence of a “canonical representation” for the universal log )) is given precisely by the class η Φ deﬁned ordinary elliptic curve (deﬁned on all of π 1 (X K by the canonical Frobenius lifting. §2. Canonical Objects Over the Stack of Multiplicative Periods In this Section, we note that by working over the stack of multiplicative periods, we can construct all the objects that we are familiar with from the case of canonical curves. The Stack of Multiplicative Periods ord Let S log = (N g,r ) log . Let Φ log be the canonical Frobenius lifting on S log (as in Chapter III, Theorem 2.8). If, for some N ≥ 1, we take the N th iterate of Φ log , we get a ﬁnite, ﬂat covering (Φ log ) N : S log → S log of S log . Let P log be the inductive limit of these coverings (as N goes to inﬁnity). Let P log be the p-adic completion of P log . Deﬁnition 2.1. We shall call P log (respectively, P log ) the universal (respectively, com- plete) stack of multiplicative periods. Unlike the rings of additive periods considered earlier, which, roughly speaking, are gen- erated by adjoining the logarithms of the multiplicative parameters to Z p , the structure sheaf of the stack of multiplicative periods is obtained essentially by adjoining all p-power roots of the multiplicative parameters to the structure sheaf of the original base scheme (or stack). More generally, let T log be a formal log scheme, whose underlying scheme T is p- adically complete and ﬂat over Z p . Let h log : Y log → T log be an r-pointed stable curve of genus g that arises from some classifying morphism T log → S log . Then if we pull back the morphism P log → S log via the classifying morphism T log → S log for h log , we obtain an object P T log → T log 167 over T log . Let P T log be the p-adic completion of P T log . We shall always assume that: log (*) The log structure of T Q is trivial over an open dense set. p For instance, typically T log will be the normalization of Z p in a ﬁnite ﬁeld extension of Q p with a log structure that is trivial in characteristic zero. Deﬁnition 2.2. We shall call P T log (respectively, P T log ) the (respectively, completed) formal scheme of multiplicative periods associated to the curve h log : Y log → T log . Note that P T log depends on the choice of classifying morphism T log → S log for the curve log that lifts the morphism T log → M g,r deﬁned by the curve itself. That is, P T log depends on a choice of quasiconformal equivalence class for h log : Y log → T log . Remark. Often in what follows we shall work in the universal case, that is, over S log , and thus obtain objects over P log (or P log ). However, one should always remember that the objects constructed deﬁne (by restriction) objects over P T log (or P T log ) for any r-pointed stable curve of genus g over T log satisfying the hypotheses just stated. The Canonical Log p-divisible Group ord log Let S log = (N g,r ) : that is, the locus of smooth ordinary curves (with a choice of quasiconformal equivalence class). Thus, the log structure on S log is trivial. Let f log : X log → S log be the universal curve. Let f log [N ] : X log [N ] → S log be the pull-back of f log by the N th iterate of the canonical Frobenius. Thus, if we pull (E, ∇ E ) back to X log [N ] and reduce modulo p N , the obstruction to deﬁning a full connection (relative to X log [N ] → Spec(A)) vanishes, and so, we obtain an MF ∇ -object, which we shall call E[N ]. Alternatively, this MF ∇ -object can be obtained taking the MF ∇ -object in Theorem 1.1, pulling back to X log [N ], reducing modulo p N , and then reducing modulo I S N . By [Falt], Theorem 7.1, away from the divisor at inﬁnity, and the method of Chapter IV, §2 (following [Kato2]) at the divisor at inﬁnity, E[N ] deﬁnes a ﬁnite, ﬂat log group object G[N ] on X log [N ]. Let P log → S log be the stack of multiplicative periods. Let f log [∞] : X log [∞] → P log be the pull-back of f log to P log . Then by restricting from X log [N ] to X log [∞] and then taking the inductive limit, we obtain an (up to ±1) log p-divisible group G[∞] = lim G[N ]| X log [∞] −→ on X log [∞]. Deﬁnition 2.3. We shall call G[∞] the canonical log p-divisible group on X log [∞]. 168 If we invert p, and pass to p-adic Tate modules, then we obtain an étale local system of Z p ⊕ Z p ’s (up to ±1) on X log [∞] Q p . Now in the notation of §1, we have Π ∞ = π 1 (X log [∞] Q p ) Thus, the p-adic Tate module of G[∞] Q p is given by the representation of Π ∞ on E 0 which was discussed in §1. Let us denote this representation by ρ ∞ : Π ∞ → GL ± (E 0 ) Deﬁnition 2.4. We shall call ρ ∞ the canonical Galois representation of Π ∞ . ord Remark. If T is any p-adically complete formal scheme which is Z p -ﬂat, and φ : T → N g,r is a morphism, then even though the canonical log p-divisible group and the canonical Galois representation are not deﬁned until one goes up to the scheme of multiplicative ord periods, one can nonetheless pull-back the canonical indigenous bundle (E, ∇ E ) N on N g,r to obtain an indigenous bundle (E, ∇ E ) T on X T log → T . This indigenous bundle is deﬁned over T , i.e., one needn’t pass to the scheme of multiplicative periods. Deﬁnition 2.5. We shall call (E, ∇ E ) T the canonical indigenous bundle on X T log . One should always remember that one only obtains the canonical indigenous bundle after ord (i.e., a quasiconformal equivalence class) of the classifying choosing a lifting φ : T → N g,r morphism T → M g,r of the curve. The Canonical Frobenius Lifting We continue with the notation of the preceding subsection. It follows from Chapter IV, Proposition 3.2, that the supersingular divisor D ⊆ X F p (where f log : X log → S log is the universal curve) is étale over S F p . We shall denote its complement in X, the ordinary locus of X log , by X ord . Now it is immediate that the construction of the canonical Frobenius lifting over the ordinary locus (preceding Theorem 1.6 of Chapter IV) carries over immediately to the present case (where the base is S, as opposed to the ring of Witt vectors with coeﬃcients in a perfect ﬁeld). Thus, if we denote base-change by Φ log (the canonical Frobenius) by means of a superscript “F ,” we obtain the universal analogue of Theorem 1.6 of Chapter IV: Theorem 2.6. There exists a unique ordinary Frobenius lifting (called canonical) log ord ) → ((X log ) ord ) F Φ log X : (X 169 over the ordinary locus such that we get a horizontal morphism Φ ∗ X E F → E which preserves the Hodge ﬁltration. Now let T be any p-adically complete formal scheme which is Z p -ﬂat, and let φ : T → ord N g,r be a morphism. Let X T log → T be the pull-back of the universal curve by φ. Write X T log F for the pull-back of the universal curve by Φ ◦ φ. Then by restricting the morphism log Φ X of Theorem 2.7, we obtain a T -morphism log ord ) T → (X log ) ord Φ log T F X T : (X Thus, in the spirit of Deﬁnition 4.18 of Chapter IV, we make the following: log Deﬁnition 2.7. We shall call X T log F the Frobenius conjugate curve to X T . We shall call log Φ log X T the p-adic Green’s function of the curve X T → T . Next, we consider compactiﬁcations of this canonical Frobenius lifting. Let D[∞] ⊆ X[∞] F p be the result of base-changing to P . Let D[∞] ⊆ X[∞] be the respective p-adic completions. Let X D [∞] be the completion of X[∞] at D[∞]. Then just as in §3 of Chapter IV, by looking at the universal deformation spaces of the canonical log p-divisible group and its double Frobenius conjugate, we obtain an isomorphism Ψ : X D F 2 [∞] ∼ = X D [∞] Let Y [∞] → X[∞] be the ﬁnite, ﬂat covering (of degree p + 1) parametrizing cyclic subgroups (in the Drinfeldian sense) of the canonical log p-divisible group. Then, just as before, there exists a divisor E ⊆ Y [∞] F p that maps isomorphically onto D[∞] F p . Let Y E [∞] be the completion of Y [∞] at E. Then we obtain an embedding (D E , R E ) : Y E [∞] → X D [∞] × P X D F 2 [∞] The image of this embedding is a divisor, which, when restricted to the ordinary locus, is equal to the union of the graph of the canonical Frobenius on (X log ) ord with its “Ψ- transpose,” as in Chapter IV, §3 (see the discussion preceding Deﬁnition 3.3). Deﬁnition 2.8. We shall call Y [∞] → X[∞], together with Ψ and (D E , R E ) the com- pactiﬁcation of the canonical Frobenius lifting Φ log X . 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USSR Sbornik 60, No. 2 (1988), 297-313. 172 Index Terminology Note: The data following each term indicate the chapter and section where the term ﬁrst appears. admissible compactiﬁed Frobenius lifting admissible Frobenius lifting admissible indigenous bundle associated Schwarz structure IV.4 IV.4 II.2 I.1 Beltrami diﬀerential Beltrami equation Bers embedding bianalytic constant function uniformizer biformal function Intro.1 Intro.1 Intro.1 I.1 I.1 I.1 I.1 canonical aﬃne parameter canonical bilinear form canonical class canonical (crystalline) Galois representation canonical curve canonical Frobenius lifting (modular) canonical Frobenius lifting (on a curve) canonical Galois extension class canonical height canonical indigenous bundle canonical lifting (point) canonical local Hecke correspondence, n th canonical local systems canonical log p-divisible group canonical multiplicative parameter III.1 II.3, III.3 III.3 IV.1, V.2 III.3 III.2 IV.1, V.2 III.1 I.2 Intro.2, V.2 III.1 IV.3 III.1, III.3 IV.2, V.2 III.1, III.3 173 canonical p-divisible group canonical pseudo-Hecke correspondence, n th canonical representation (complex case) canonical uniformization compactiﬁcation of a Frobenius lifting compactiﬁcation of the canonical Frobenius lifting crystalline-induced Galois representation crystalline induction crystalline Schwarz structure with nilpotent monodromy III.1 IV.3 Intro.1, Intro.2 III.3 IV.4 V.2 V.1 V.0 I.1 I.1 diﬀerential local system dilatation divisor of marked points III.1, III.3 Intro.1 I.1 elliptic Riemann surface Intro.1 Fenchel-Nielsen coordinates ﬁnite type (Riemann surface of) F L-bundle Frobenius conjugate curve Frobenius lifting Frobenius on R 1 f ∗ τ X log /S log Intro.1 Intro.1 II.1 V.2 III.1 II.2 generic uniformization number global n th canonical Hecke correspondence Green’s function (complex case) Green’s function, p-adic II.3 IV.3 Intro.1 IV.4, V.2 Hasse invariant Hecke operators Hecke type height of a Frobenius lifting Hilbert transformation II.2, II.3 IV.3 IV.3 IV.4 Intro.1 174 Hodge section hyperbolic metric hyperbolic Riemann surface hyperbolic volume I.2 Intro.1 Intro.1 Intro.1 Igusa curve indigenous bundle of restrictable type indigenous section induced MF ∇ -object inﬁnitesimal Verschiebung intrinsic bundle II.3 Intro.2, I.2 III.2 V.1 II.2 I.2 Jacobson’s formula II.2 Kähler metrics and Frobenius liftings Kodaira-Spencer morphism Intro.2 I.1, I.2 local compactiﬁcation of a Frobenius lifting local height of a Frobenius lifting locally intrinsic bundle locally stable of dimension one log admissible covering log p-divisible group IV.4 IV.4 I.2 I.1 III.2 IV.2 marked MF ∇ -object I.1 II.2 naive compactiﬁcation of a Frobenius lifting nilpotent indigenous bundle normalized P 1 -bundle with connection IV.4 II.2 I.1 175 I.2 ordinary curve, hyperbolically parabolically ordinary Frobenius lifting ordinary indigenous bundle ordinary locus II.3 II.3 III.1 II.3 IV.1 p-curvature parabolic Riemann surface parabolic structure parabolic volume element partition curves pre-Schwarz structure pseudo-uniformizer II.1 Intro.1 I.1 Intro.1 Intro.1 I.1 I.1 quasiconformal equivalence class, p-adic quasiconformal function quasidisk III.3 Intro.1 Intro.1 real Riemann surface renormalized Frobenius pull-back ring of additive periods Intro.1 III.2 III.1 scheme of multiplicative periods Schwarz structure Schwarzian derivative Serre-Tate parameter singular small aﬃne small parameters smooth and unmarked stable curve stack of multiplicative periods supersingular divisor V.2 I.1 I.1 IV.1 I.1 V.1 V.1 I.1 I.2 V.2 II.2 176 tangential local system Teichmüller metric Teichmüller uniformization topological marking, p-adic totally degenerate curve trianalytic constant function triformal function III.1, III.3 Intro.1 Intro.1 III.3 I.3 I.1 I.1 I.1 uniformizing Galois representation uniformizing MF ∇ -object III.1 III.1 Verschiebung of an indigenous bundle log ⊗2 ) (−D) on f ∗ (ω X/S morphism II.2 II.2 II.2 Weil-Petersson inner product Weil-Petersson metric Wolpert’s coordinates of degeneration Intro.1 Intro.1 Intro.1 Major Notation A Φ B B μ B + (−) C III.1 Intro.1, II.3 III.3 V.1 I.2, IV.4 (D, ∇ D ) Δ D E (D E , R E ) III.2 V.1 II.2 V.2 177 d S D S ; I S D S Gal (D y , R y ) D 0 (a, b) (E D , ∇ E D ) E E Gal E X (E N , ∇ E N ) η Φ E 0 F ∗ I.1 V.1 V.1 IV.3 IV.2 V.1 II.2 V.1 III.2 III.1 V.1 II.2, III.2 G g,r ; G Δ G log GL ± G[∞] H E H Y log ht(Φ); ht x (Φ) H x ι (a,b);0 II.3 IV.2 I.1 V.2 II.2 IV.3 IV.4 IV.3 IV.2 J κ κ A κ β L S ; L × S I.1 V.1 III.1 III.1, III.3 I.1 M g,r I.2, ord M g,r μ = (Δ, ) N g,r ord N g,r Ω et g,r Ω et N Ω et Φ Ω log S III.2 II.3 III.3 II.2 II.3, III.2 II.3 III.3 III.1 V.1 178 × ω U log /S log I.1 O U # I.1 P (P, F · (P), ∇ P , P Φ ) P E P et Φ E Φ log N Φ log X Φ ω E Φ τ E Π 1 ; Π ∞ P log Ψ Ψ αβ Ψ x Q Q g,r q ω,α q ω,z q θ ρ ρ ∞ II.1 III.1 II.2 III.1 IV.1, V.1 III.2 IV.1, V.2 II.2 II.2 V.1 III.3 V.2 V.2 III.1 IV.3 Intro.1 II.2 III.3 II.1 IV.1 IV.1 V.2 S; S × S g,r I.1 I.3 adm S g,r T T 1/2 Θ E Θ et g,r Θ et N Θ et Φ Θ F ∗ Θ V E U bi U bi U can II.2 II.1, IV.1 IV.1 II.2 II.3 III.3 III.1 III.2 II.2 I.1 I.1 III.1, III.3 179 U tr U tr I.1 I.1 V E V g,r V μ,β X(a, b) ord X ord ; (X log ) ord X 0 (n) II.2 II.2 III.3 IV.3 IV.1 IV.2 180